Question 1

A natural number, when divided by each of 9, 10, 12, and 15, leaves a remainder of 3 in every case. Find the smallest such natural number that satisfies all these remainder conditions simultaneously.

Accepted Answer

Introduction / Context: This problem involves congruences and least common multiples. The condition that a number leaves the same remainder when divided by several divisors can be rephrased in terms of a number being a fixed amount greater than a common multiple of those divisors. This is a typical pattern in modular arithmetic questions that appear in competitive exams. Given Data / Assumptions: Let the required number be N. N leaves a remainder of 3 when divided by 9, 10, 12, and 15. We want the smallest natural number N satisfying all these conditions. Concept / Approach: If N leaves remainder 3 when divided by each of the given divisors, then N minus 3 is exactly divisible by 9, 10, 12, and 15. In other words, N - 3 is a common multiple of these numbers. The smallest such N corresponds to N - 3 being the least common multiple (LCM) of these divisors, and then we add 3 back to find N. Step-by-Step Solution: Compute LCM of 9, 10, 12, and 15. 9 = 3^2 10 = 2 * 5 12 = 2^2 * 3 15 = 3 * 5 Take highest powers: 2^2, 3^2, 5. LCM = 2^2 * 3^2 * 5 = 4 * 9 * 5 = 180. So N - 3 must be a multiple of 180. For the smallest N, take N - 3 = 180, so N = 180 + 3 = 183. Verification / Alternative check: Check N = 183: 183 / 9 gives remainder 3. 183 / 10 gives remainder 3. 183 / 12 gives remainder 3. 183 / 15 gives remainder 3. Thus 183 satisfies all conditions. Any smaller number would correspond to N - 3 being a smaller common multiple than 180, which is impossible since 180 is the least common multiple. Why Other Options Are Wrong: 153, 63, 123, and 303 either fail to give remainder 3 with one or more of the divisors or are not the smallest such number. For example, 153 divided by 10 does not leave remainder 3, and 303 is larger than the minimal solution 183. Common Pitfalls: A frequent mistake is to misinterpret the condition and look for a number divisible by the divisors instead of one leaving a fixed remainder. Another common issue is computing the LCM incorrectly by not taking the highest powers of primes. Always convert remainder conditions to a common multiple condition by subtracting the remainder, then use the LCM to find the smallest solution. Final Answer: 183

Question 2

The product of two two digit natural numbers is 2160 and their highest common factor (HCF) is 12. Using this information, determine which pair of numbers among the options represents the correct two numbers.

Accepted Answer

Introduction / Context: This question combines the use of product, highest common factor (HCF), and simple factorization to determine two unknown numbers. Knowing the product and the HCF provides a strong structure for working backwards. Such questions are common in number system and arithmetic sections of competitive exams. Given Data / Assumptions: The two numbers are two digit natural numbers. Their product is 2160. Their HCF is 12. We need to find the correct ordered pair from the given options. Concept / Approach: If the HCF of two numbers is 12, we can represent them as 12x and 12y, where x and y are co prime integers (their HCF is 1). The product of the two numbers is then 12 * 12 * x * y = 144xy. Since this product is given as 2160, we can solve for xy and then find co prime factor pairs of that value that produce two digit numbers 12x and 12y. Step-by-Step Solution: Let the numbers be 12x and 12y. Given product: 12x * 12y = 2160. So 144xy = 2160. Therefore xy = 2160 / 144 = 15. Now find co prime factor pairs of 15: (1, 15) and (3, 5). Corresponding number pairs: (12 * 1, 12 * 15) = (12, 180) and (12 * 3, 12 * 5) = (36, 60). Only (36, 60) gives two digit numbers for both entries. Verification / Alternative check: Check HCF and product: HCF(36, 60) = 12. Product: 36 * 60 = 2160. The pair (36, 60) satisfies both conditions exactly. No other candidate pair among the options matches both the HCF and product constraints with both numbers being two digit. Why Other Options Are Wrong: (12, 60) has product 720, not 2160, and 12 is not a two digit number. (72, 30) has product 2160 but HCF is 6, not 12. (60, 72) also has product 4320, which is not equal to 2160, and its HCF is 12 but the product is incorrect. (18, 120) includes a three digit number and its HCF and product do not match the required values. Common Pitfalls: Many students forget that the co factors x and y must be co prime when the HCF is already factored out. Another mistake is not enforcing the two digit constraint or miscalculating the product. Always express numbers in the form HCF times co prime factors and use the product to determine these co factors. Final Answer: (36, 60)

Question 3

Among the numbers 99, 101, 176, and 182, which one has the greatest number of positive divisors? Use prime factorization to compare the total count of divisors for each option.

Accepted Answer

Introduction / Context: This question tests understanding of how the number of divisors of an integer is related to its prime factorization. Instead of calculating all divisors manually, we use a formula based on the exponents of primes in the factorization. Comparing several numbers in this way is an efficient technique for identifying which has the most divisors. Given Data / Assumptions: Numbers to compare: 99, 101, 176, 182. We want to find which number has the greatest number of positive divisors. Concept / Approach: If a positive integer n has prime factorization n = p1^a1 * p2^a2 * ... * pk^ak, then the total number of positive divisors of n is (a1 + 1) * (a2 + 1) * ... * (ak + 1). Therefore, we factor each candidate number, compute this divisor count, and then compare the results to see which is largest. Step-by-Step Solution: Factor 99: 99 = 3^2 * 11^1. Number of divisors of 99 = (2 + 1) * (1 + 1) = 3 * 2 = 6. Factor 101: 101 is a prime, so 101 = 101^1. Number of divisors of 101 = (1 + 1) = 2. Factor 176: 176 = 2^4 * 11^1. Number of divisors of 176 = (4 + 1) * (1 + 1) = 5 * 2 = 10. Factor 182: 182 = 2^1 * 7^1 * 13^1. Number of divisors of 182 = (1 + 1) * (1 + 1) * (1 + 1) = 2 * 2 * 2 = 8. Compare counts: 6, 2, 10, 8. The maximum is 10 for 176. Verification / Alternative check: We can list a few divisors of 176 to see that it has many: 1, 2, 4, 8, 11, 16, 22, 44, 88, 176. This matches the count of 10 divisors computed from the formula. The other numbers have fewer divisors based on the exponent method, so the result is consistent. Why Other Options Are Wrong: 99 has 6 divisors, 101 has 2 divisors, and 182 has 8 divisors, all of which are fewer than the 10 divisors of 176. 90 is not in the original comparison set and is only included as a distractor. Common Pitfalls: Learners sometimes try to list divisors exhaustively for each number, which is time consuming and error prone. Others may incorrectly factor a number and then misapply the formula. Carefully factor each number into primes and use the exponent plus one rule for each prime factor to get the correct divisor count efficiently. Final Answer: 176

Question 4

The least common multiple (LCM) and highest common factor (HCF) of two numbers are 4284 and 32 respectively. If one of the numbers is 672, find the other number using the relationship between LCM, HCF, and the product of two numbers.

Accepted Answer

Introduction / Context: This question tests a standard and very useful identity connecting two numbers, their highest common factor (HCF), and their least common multiple (LCM). The product of two positive integers equals the product of their HCF and LCM. Using this identity, we can find an unknown number when the other number, the HCF, and the LCM are known. Given Data / Assumptions: LCM of the two numbers = 4284 HCF of the two numbers = 32 One number = 672 Both numbers are positive integers. Concept / Approach: For any two positive integers a and b, the fundamental relationship is: a * b = HCF(a, b) * LCM(a, b). If one number, the HCF, and the LCM are known, we can rearrange this to find the other number. Specifically, if a is known and b is unknown, then: b = (HCF * LCM) / a. Step-by-Step Solution: Let the unknown number be x. Given: LCM = 4284, HCF = 32, known number = 672. Use the identity: 672 * x = 32 * 4284. Compute the right side: 32 * 4284 = 137088. So 672 * x = 137088. Solve for x: x = 137088 / 672. Perform division: 137088 / 672 = 204. Therefore, the second number is 204. Verification / Alternative check: We can verify by recomputing HCF and LCM of 672 and 204. The HCF of 672 and 204 is 32, and the LCM is 4284. Also, 672 * 204 = 137088, which equals 32 * 4284, confirming that the identity holds and that the calculated number is correct. Why Other Options Are Wrong: 102, 64, 92, and 156 do not satisfy the identity when paired with 672. For each of these, 672 multiplied by the option either does not equal 32 multiplied by 4284 or leads to inconsistent HCF and LCM values. Common Pitfalls: A common error is to add or subtract the LCM and HCF instead of using the product identity. Another mistake is to perform the division incorrectly or to forget that the HCF and LCM relationship applies only to pairs of numbers. Carefully applying the formula a * b = HCF * LCM prevents such mistakes. Final Answer: 204

Question 5

If a and b are two co prime positive integers with a greater than b, what is the least common multiple (LCM) of a and b expressed in terms of a and b?

Accepted Answer

Introduction / Context: This conceptual question focuses on the properties of co prime numbers. Two numbers are co prime if their highest common factor (HCF) is 1. The relationship between the product of two numbers, their HCF, and their least common multiple (LCM) simplifies significantly when the numbers are co prime. Understanding this is important for solving many number theory and arithmetic problems quickly. Given Data / Assumptions: a and b are positive integers. a and b are co prime, which means HCF(a, b) = 1. a is greater than b. We need the LCM of a and b in algebraic form. Concept / Approach: For any two positive integers a and b, the fundamental identity is: a * b = HCF(a, b) * LCM(a, b). If a and b are co prime, then HCF(a, b) = 1. Substituting this into the identity simplifies the relationship, making LCM(a, b) equal to the product a * b. Step-by-Step Solution: Given HCF(a, b) = 1 because a and b are co prime. Using the identity: a * b = HCF(a, b) * LCM(a, b). This becomes a * b = 1 * LCM(a, b). Therefore, LCM(a, b) = a * b. So the LCM of two co prime numbers a and b is ab. Verification / Alternative check: Consider simple co prime examples: Example 1: a = 3, b = 4. LCM(3, 4) = 12, and ab = 3 * 4 = 12. Example 2: a = 5, b = 7. LCM(5, 7) = 35, and ab = 5 * 7 = 35. In each case, the LCM equals the product of the numbers, confirming the general result for co prime pairs. Why Other Options Are Wrong: a or b alone cannot be the LCM because the LCM must be divisible by both numbers, and a single number will not be divisible by the other unless they are equal, which need not be true. a + b and a - b generally do not satisfy the divisibility conditions required for the LCM, except in special accidental cases. They are not guaranteed formulas for all co prime pairs. Common Pitfalls: Some learners confuse co prime numbers with consecutive numbers or with numbers having a small HCF. Others may forget the product identity and try to guess based on the size of expressions. Remember that co prime means HCF equals 1, and this makes the LCM equal to the plain product a * b by the standard identity. Final Answer: ab

Question 6

The sum of two natural numbers is 55. Their highest common factor (HCF) is 5 and their least common multiple (LCM) is 120. Find the sum of the reciprocals of the two numbers in simplest fractional form.

Accepted Answer

Introduction / Context: This question cleverly combines the ideas of HCF, LCM, and reciprocals of numbers. Instead of directly finding the numbers, we can use the relationship between HCF, LCM, and the product of two numbers to compute the sum of their reciprocals in a quick and elegant way. This type of reasoning is useful in higher level arithmetic and algebra problems. Given Data / Assumptions: Let the numbers be x and y. x + y = 55. HCF(x, y) = 5. LCM(x, y) = 120. We need the value of 1/x + 1/y. Concept / Approach: For two numbers x and y, there is a well known identity: x * y = HCF(x, y) * LCM(x, y). Once we know the product x * y and the sum x + y, we can find the sum of reciprocals easily: 1/x + 1/y = (x + y) / (x * y). We therefore compute the product from the HCF and LCM, and then form the simple fraction with the given sum. Step-by-Step Solution: Given HCF = 5 and LCM = 120. Compute product: x * y = 5 * 120 = 600. Given sum: x + y = 55. Sum of reciprocals: 1/x + 1/y = (x + y) / (x * y). Substitute values: 1/x + 1/y = 55 / 600. Simplify the fraction: divide numerator and denominator by 5. 55 / 600 = 11 / 120. Therefore, the sum of the reciprocals is 11/120. Verification / Alternative check: We can go further and find x and y explicitly to check. Since HCF is 5, let x = 5a and y = 5b, with HCF(a, b) = 1. The product x * y = 600 implies 25ab = 600, so ab = 24. Also, x + y = 5a + 5b = 55, so a + b = 11. Co prime factors of 24 that sum to 11 are a = 3 and b = 8. Then x = 15 and y = 40. Their reciprocals are 1/15 and 1/40. Sum = (8 + 3) / 120 = 11 / 120, confirming our earlier result. Why Other Options Are Wrong: 55/601, 601/55, 120/11, and 5/120 do not match the correctly derived expression (x + y) / (x * y). They may look similar or inverted, but they do not come from the established product and sum relationship. Common Pitfalls: Some learners attempt to guess x and y without using the HCF and LCM product identity, which can be slow and error prone. Others mistakenly add HCF and LCM or confuse 1/x + 1/y with 1/(x + y). Remember that the key formula for reciprocals is (x + y) / (x * y) and that the product can be obtained directly from HCF and LCM for two numbers. Final Answer: 11/120

Question 7

A student can arrange her collection of books into equal groups of 5 books, 9 books, or 13 books each, with no book left over in any case. What is the smallest possible total number of books that the student can have?

Accepted Answer

Introduction / Context: This aptitude question tests understanding of the Least Common Multiple, or LCM, which is a common concept in number system problems. When a quantity can be grouped in several different ways without leaving any remainder, it means the total quantity is exactly divisible by all the given group sizes. The smallest such total is the LCM of the given numbers. Understanding LCM is very useful in scheduling, grouping, and divisibility based questions that often appear in competitive exams. Given Data / Assumptions: The student can arrange books in groups of 5. The student can arrange books in groups of 9. The student can arrange books in groups of 13. No book is left over in any of these arrangements. We need the smallest possible total number of books. Concept / Approach: To find the smallest positive number that is divisible by 5, 9, and 13, we calculate the LCM of these three numbers. The LCM is found by taking prime factorizations and choosing the highest power of each prime involved. The resulting LCM represents the smallest number that can be divided exactly by all the given divisors, which in this context corresponds to the smallest possible number of books. Step-by-Step Solution: Step 1: Prime factorize each number. 5 = 5 9 = 3 * 3 = 3^2 13 = 13 Step 2: Collect highest powers of primes. We have primes 3, 5, and 13. LCM = 3^2 * 5 * 13 Step 3: Multiply to get the LCM. 3^2 = 9 9 * 5 = 45 45 * 13 = 585 Therefore, the smallest total number of books is 585. Verification / Alternative check: We verify by dividing 585 by each group size. 585 / 5 = 117, 585 / 9 = 65, and 585 / 13 = 45. All quotients are integers, and there is no remainder in any case. This confirms that 585 is divisible by 5, 9, and 13, and hence satisfies the condition of forming equal groups for all three sizes. Because LCM gives the smallest such number, 585 is indeed the minimum possible total number of books. Why Other Options Are Wrong: 487: Not divisible by 5 or 9, so it cannot form groups of those sizes without remainder. 635: Divisible by 5 but not divisible by 9 or 13 together, so it fails the requirement. 705: Divisible by 5 and 9 but not exactly by 13, so it is not valid for all group sizes. 845: Divisible by 5 and 13 but not by 9, hence it does not satisfy all the conditions. Common Pitfalls: Many students mistakenly try to test numbers one by one or simply multiply all three numbers without using LCM logic. Another common mistake is to confuse HCF (the greatest common divisor) with LCM. Some examinees also forget to include the highest power of a prime factor when finding the LCM, leading to an incorrect smaller value. Remember that LCM must be divisible by each number completely, without remainder. Final Answer: The smallest possible number of books is 585.

Question 8

Find the Highest Common Factor (HCF) of the two numbers 3341 and 3328.

Accepted Answer

Introduction / Context: This problem focuses on finding the Highest Common Factor, or HCF, of two given numbers. The HCF is the largest positive integer that divides both numbers exactly, with no remainder. Questions of this type measure understanding of divisibility, factors, and sometimes the use of simple techniques like taking differences. HCF problems are fundamental in number theory and very important for competitive exams and school level aptitude tests. Given Data / Assumptions: First number = 3341. Second number = 3328. We need to determine the HCF of these two numbers. No additional hidden conditions are present. Concept / Approach: When two numbers are close to each other, a very efficient approach to find their HCF is to consider their difference. Any common factor of the two numbers must also divide their difference. Once the difference is found, we test if this difference divides both numbers exactly. If it does, that difference is a strong candidate for the HCF. We can then confirm by checking divisibility and ensuring that no larger common factor exists. Step-by-Step Solution: Step 1: Compute the difference between the two numbers. Difference = 3341 - 3328 = 13 Step 2: Check if 13 divides both numbers exactly. 3341 / 13 = 257 (an integer) 3328 / 13 = 256 (an integer) Step 3: Since 13 divides both numbers exactly, it is a common factor. Step 4: Because the numbers are close, and 13 is already a significant prime factor, there is no larger common factor than 13 that divides both. Therefore, HCF(3341, 3328) = 13. Verification / Alternative check: An alternative approach is to use the Euclidean algorithm. We repeatedly take remainders until we reach zero. Starting with 3341 and 3328, the difference based method we used is essentially a simplified Euclidean approach. Because we found that the difference is 13 and it divides both numbers exactly, and no larger candidate exists, the result is confirmed. This gives us confidence that 13 is indeed the highest common factor of the two numbers. Why Other Options Are Wrong: 31: Does not divide both 3341 and 3328 exactly, so it cannot be the HCF. 257: Divides 3341, but 3328 / 257 is not an integer, so it is not a common factor. 337: Divides neither of the numbers exactly, so it is invalid. 1: While 1 is always a common factor, it is not the highest common factor in this case because we have already found 13 as a larger common divisor. Common Pitfalls: A typical mistake is to guess factors without checking divisibility carefully or to assume that because numbers are large, the HCF must also be large. Another mistake is ignoring the difference method, which is especially helpful when numbers are close. Some learners also confuse HCF with LCM and may try to multiply factors instead of looking for common divisors. Always remember that HCF is about the greatest number that divides all given numbers exactly. Final Answer: The HCF of 3341 and 3328 is 13.

Question 9

The product of two positive integers is 3026 and their Least Common Multiple (LCM) is 89. Using the relationship between HCF and LCM, find the Highest Common Factor (HCF) of the two numbers.

Accepted Answer

Introduction / Context: This question directly tests knowledge of the fundamental relationship between two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). Many competitive exams include problems where either the product, HCF, or LCM is missing, and you are expected to use a standard formula to find the unknown term. Mastering this relationship makes it easier to handle a large variety of number system questions quickly and accurately. Given Data / Assumptions: Product of the two numbers = 3026. LCM of the two numbers = 89. The numbers are positive integers. We need to find the HCF of the two numbers. Concept / Approach: For any two positive integers a and b, there is a well known relationship: a * b = HCF(a, b) * LCM(a, b). This formula connects the product of the numbers with their HCF and LCM. When any two of these three quantities are known, the third can be computed easily. In this problem, we know the product and the LCM, so we can rearrange the formula to find the HCF as: HCF = (product of the numbers) / (LCM). Step-by-Step Solution: Step 1: Write the formula. a * b = HCF * LCM Step 2: Substitute the known values. 3026 = HCF * 89 Step 3: Solve for HCF. HCF = 3026 / 89 Step 4: Perform the division. 3026 / 89 = 34 Therefore, the HCF of the two numbers is 34. Verification / Alternative check: To verify, we can assume that the two numbers are HCF * m and HCF * n, where m and n are co-prime and their product equals the LCM divided by HCF relationships. However, the quickest verification is simply to multiply the found HCF and the given LCM: 34 * 89. This equals 3026, which matches the original product. This confirms that the calculation is correct and that 34 is the correct HCF. Why Other Options Are Wrong: 33: If HCF were 33, then HCF * LCM = 33 * 89 = 2937, which does not equal 3026. 35: 35 * 89 = 3115, which does not match the given product. 29: 29 * 89 = 2581, again not equal to 3026. 17: 17 * 89 = 1513, which is exactly half of 3026, so it cannot be the HCF when using the given data. Common Pitfalls: Learners sometimes try to factorize the product directly into two numbers and then find their HCF and LCM manually. While possible, that approach is slower and more error prone in a timed exam. Another common mistake is to invert the formula or confuse HCF with LCM. Always remember the clean relationship: product of the numbers equals HCF multiplied by LCM. Using this formula avoids unnecessary effort and leads directly to the answer. Final Answer: The HCF of the two numbers is 34.

Question 10

Consider any set of positive integers. The Least Common Multiple (LCM) of these numbers is related to the product of all the numbers. Which statement best describes the minimum possible value of the LCM compared to the product of the numbers?

Accepted Answer

Introduction / Context: This is a conceptual question about the relationship between the Least Common Multiple (LCM) of a set of numbers and the product of those numbers. Rather than asking you to compute any specific numerical value, the question focuses on understanding how large or small the LCM can be when compared with the product. Concept questions like this are common in number system topics, because they help build intuition that can later be applied to trickier word problems and algebraic situations involving divisibility and multiples. Given Data / Assumptions: We are given an arbitrary set of positive integers. LCM refers to the least positive integer that is divisible by each number in the set. The product refers to the result of multiplying all the numbers together. We are asked about the minimum possible value of the LCM relative to the product. All numbers are assumed to be non zero positive integers. Concept / Approach: The product of a set of positive integers includes every prime factor of each number, including repetitions. In contrast, the LCM uses the highest power of each distinct prime factor appearing in any of the numbers. This means that the LCM never has more prime factors (counted with multiplicity) than the product. As a result, the LCM is always less than or equal to the product. If the numbers are co prime, the LCM happens to be exactly equal to the product. If they share common factors, the LCM is strictly less than the product because repeated prime factors are not counted multiple times in the LCM. Step-by-Step Solution: Step 1: Recall definition of product. Product = n1 * n2 * n3 * ... * nk. Step 2: Recall definition of LCM. LCM = smallest positive integer divisible by all n1, n2, ..., nk. Step 3: Express each number in prime factor form. Each number is written as a product of primes raised to some powers. Step 4: For product, multiply all prime powers from all numbers. Step 5: For LCM, take only the highest power of each distinct prime. Step 6: Because LCM avoids unnecessary repetitions, it can never exceed the product and is therefore always less than or equal to the product. Verification / Alternative check: Consider a simple example with co prime numbers such as 2 and 3. Their product is 2 * 3 = 6. Their LCM is also 6 because 6 is the smallest number divisible by both 2 and 3. In this case, LCM equals product, which fits the statement that LCM is less than or equal to the product. Now take numbers that share a common factor, for example 4 and 6. The product is 4 * 6 = 24. The LCM is 12, which is clearly less than 24. Again, this supports the idea that LCM is less than or equal to the product, and sometimes strictly less. Why Other Options Are Wrong: The LCM is always greater than the product: This is false, because we just saw examples where LCM equals or is less than the product. The LCM is always equal to the product: This is not true when numbers share common factors, such as 4 and 6, where LCM is 12 but product is 24. The relationship between LCM and product cannot be determined: This is incorrect because number theory gives a clear relationship: LCM is always less than or equal to the product. The LCM is always less than each individual number: This is false since LCM is at least as large as the largest of the given numbers, and often much larger. Common Pitfalls: Many learners confuse LCM with HCF and assume similar size relations without properly analyzing the definitions. Another frequent mistake is to think that LCM must always be equal to the product, mainly because in simple examples with co prime numbers that pattern holds. However, once numbers share factors, this assumption breaks down. Also, some students think the relationship is unpredictable, but in reality the prime factor method gives a very systematic way to understand the size of the LCM relative to the product. Final Answer: The minimum possible value of the LCM is always less than or equal to the product of all the given numbers.

Question 11

The Highest Common Factor (HCF) of two numbers is 11 and their Least Common Multiple (LCM) is 330. If one of the numbers is 55, find the other number using the relationship between HCF, LCM, and the product of the numbers.

Accepted Answer

Introduction / Context: This question checks whether you know how to connect HCF, LCM, and the product of two numbers. Such problems are standard in number system topics in aptitude exams. By using a single formula, we can move quickly from given values to the unknown number without trial and error. Understanding this relationship is not only useful for direct HCF and LCM questions but also for many mixture and ratio problems where similar reasoning appears in disguised forms. Given Data / Assumptions: HCF of two numbers = 11. LCM of the two numbers = 330. One of the numbers = 55. Both numbers are positive integers. We need to find the other number. Concept / Approach: For any two positive integers a and b, the relationship between HCF and LCM is: a * b = HCF(a, b) * LCM(a, b). If we know one number, the HCF, and the LCM, we can rearrange this formula to find the second number. Specifically, other number = (HCF * LCM) / known number. This formula is very efficient and avoids the need to factorize the numbers manually. Step-by-Step Solution: Step 1: Write the standard formula. a * b = HCF * LCM Step 2: Substitute the known values into the formula. Let the other number be x. 55 * x = 11 * 330 Step 3: Calculate the right hand side. 11 * 330 = 3630 Step 4: Solve for x. x = 3630 / 55 Step 5: Perform the division. 3630 / 55 = 66 Therefore, the other number is 66. Verification / Alternative check: We can verify the answer by checking both the HCF and LCM. The two numbers are 55 and 66. Their HCF is 11 because both are divisible by 11 and no higher common factor exists. To check the LCM, we can use the formula again: LCM = (55 * 66) / 11 = 3630 / 11 = 330. This matches the given LCM, so the calculation is consistent. Hence the result is correct. Why Other Options Are Wrong: 33: If the other number were 33, then the product would be 55 * 33 = 1815, leading to an incorrect LCM when combined with HCF 11. 44: With 44, the product is 55 * 44 = 2420; dividing by 11 gives LCM = 220, not 330. 77: With 77, product is 55 * 77 = 4235; dividing by 11 gives LCM = 385, again not equal to 330. 110: Here, product 55 * 110 = 6050, and 6050 / 11 = 550, which does not match the given LCM. Common Pitfalls: A common error is to try to guess the other number by random factorization or by relying only on HCF or only on LCM instead of using both together. Another mistake is to misapply the formula and write HCF * LCM = HCF + LCM or some other incorrect combination. Some students also forget to simplify the division correctly, leading to calculation mistakes. Always write the formula clearly, substitute carefully, and compute step by step. Final Answer: The other number is 66.

Question 12

If the product of two positive integers is 4941 and their Least Common Multiple (LCM) is 81, find the Highest Common Factor (HCF) of the two numbers.

Accepted Answer

Introduction / Context: This question again applies the relationship between the product of two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). Such relationships are a standard part of quantitative aptitude syllabi. When the product and LCM are known, the HCF can be obtained directly using a simple division. Understanding this saves a lot of time when dealing with large numbers where manual factorization is not practical. Given Data / Assumptions: Product of the two numbers = 4941. LCM of the two numbers = 81. The numbers are positive integers. We need to find the HCF of the two numbers. Concept / Approach: The standard formula for any two positive integers a and b is: a * b = HCF(a, b) * LCM(a, b). We can use this to find one unknown when the other three quantities are known. In this problem, we know the product and the LCM, and we want the HCF. Rearranging the formula, we get: HCF = (product of the numbers) / (LCM). This is a direct and efficient method especially when the product is exactly divisible by the LCM, as it should be for valid integers. Step-by-Step Solution: Step 1: Write the relationship formula. a * b = HCF * LCM Step 2: Substitute given values. 4941 = HCF * 81 Step 3: Solve for HCF. HCF = 4941 / 81 Step 4: Perform the division. 4941 / 81 = 61 Therefore, the HCF of the two numbers is 61. Verification / Alternative check: We can perform a quick verification by multiplying the obtained HCF and the given LCM: 61 * 81. Calculating this gives 61 * 81 = 4941, which matches the given product exactly. This confirms that the values are consistent with the fundamental formula. If the multiplication did not give the original product, it would mean either the division was done incorrectly or some misunderstanding of the relationship occurred. Why Other Options Are Wrong: 60: 60 * 81 = 4860, which is less than 4941, so 60 cannot be the HCF in this situation. 59: 59 * 81 = 4779, which does not match the product 4941. 35: 35 * 81 = 2835, far from the required product, so this is incorrect. 27: 27 * 81 = 2187, again very different from the given product, so it cannot be right. Common Pitfalls: One common mistake is to forget the exact formula and mix up HCF and LCM or attempt to factorize the product into two numbers directly, which can be time consuming. Another pitfall is to perform the division incorrectly, especially when working without a calculator. It is helpful to check the final result by multiplying HCF and LCM back to see if the original product is recovered. This extra verification step is quick but very effective in avoiding silly errors in exams. Final Answer: The HCF of the two numbers is 61.

Question 13

Two numbers are in the ratio 3 : 5 and their Highest Common Factor (HCF) is 20. Using this information, find the Least Common Multiple (LCM) of the two numbers.

Accepted Answer

Introduction / Context: This question combines the idea of ratios with the concepts of HCF and LCM. Problems like this are common in aptitude tests because they check whether a student can move from a ratio to actual numbers and then apply number theory formulas. Once we express the two numbers in terms of their ratio and HCF, it becomes straightforward to calculate the Least Common Multiple, which is required here. Given Data / Assumptions: The ratio of the two numbers is 3 : 5. HCF of the two numbers = 20. We need to find the LCM of the two numbers. Both numbers are positive integers. Concept / Approach: If two numbers are in the ratio m : n and their HCF is h, then the numbers can be expressed as: first number = 3 * 20 second number = 5 * 20 more generally, number1 = h * m and number2 = h * n. Once we know the explicit values of both numbers, we can find the LCM using either prime factorization or the relationship: LCM = (product of the numbers) / HCF. This is often the quickest approach for two numbers when HCF is known. Step-by-Step Solution: Step 1: Use the ratio and HCF to find the actual numbers. Ratio = 3 : 5, HCF = 20. Number 1 = 3 * 20 = 60. Number 2 = 5 * 20 = 100. Step 2: Use the relationship between product, HCF, and LCM. Number 1 * Number 2 = HCF * LCM. 60 * 100 = 20 * LCM. 6000 = 20 * LCM. LCM = 6000 / 20 = 300. Therefore, the LCM of the two numbers is 300. Verification / Alternative check: To verify, we can compute the LCM by prime factorization. The numbers are 60 and 100. Prime factorization gives: 60 = 2^2 * 3 * 5, 100 = 2^2 * 5^2. The LCM takes the highest powers of each prime: 2^2, 3, and 5^2. Thus, LCM = 2^2 * 3 * 5^2 = 4 * 3 * 25 = 300. This matches the value found using the relationship formula, confirming that 300 is correct. Why Other Options Are Wrong: 30: This is far smaller than both numbers and cannot be their LCM since LCM must be at least as large as the largest number, which is 100. 60: While 60 is one of the numbers, it does not evenly divide 100 in a way that makes it the least common multiple of both 60 and 100. 10: This is actually a factor of both numbers, but it is a common factor rather than the least common multiple. 240: Although 240 is a common multiple of 60, it is not divisible by 100, so it cannot be the LCM of 60 and 100. Common Pitfalls: A common mistake is to treat the HCF as if it were one of the numbers directly or to misapply the ratio without multiplying by the HCF. Another error is forgetting the relationship LCM = (product) / HCF and instead attempting to list multiples manually, which is inefficient. Keeping the connection between ratio, HCF, and actual numbers clear makes these problems straightforward to solve. Final Answer: The LCM of the two numbers is 300.

Question 14

The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers are 11 and 825 respectively. If one of the numbers is 275, find the other number.

Accepted Answer

Introduction / Context: This question is another application of the key relationship between the product of two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). Such problems appear frequently in competitive exams, and solving them efficiently is a great way to save time. Once you remember the core formula, the calculation is direct, even if the numbers involved seem large at first glance. Given Data / Assumptions: HCF of the two numbers = 11. LCM of the two numbers = 825. One of the numbers = 275. We need to find the other number. The numbers are positive integers. Concept / Approach: The fundamental formula for any two positive integers a and b is: a * b = HCF(a, b) * LCM(a, b). Here, HCF and LCM are provided, and one of the numbers is also given. We can simply rearrange the formula to find the missing second number. Specifically: other number = (HCF * LCM) / known number. This avoids manual factorization and provides a neat direct route to the answer. Step-by-Step Solution: Step 1: Write the formula. a * b = HCF * LCM Step 2: Let the other number be x. 275 * x = 11 * 825 Step 3: Multiply HCF and LCM. 11 * 825 = 9075 Step 4: Solve for x by dividing. x = 9075 / 275 Step 5: Simplify the division. 9075 / 275 = 33 Therefore, the other number is 33. Verification / Alternative check: We can verify by checking the HCF and LCM of 275 and 33. First, note that 275 = 11 * 25 and 33 = 11 * 3. The common factor is 11, so HCF = 11. Next, multiply the numbers: 275 * 33 = 9075. Using the formula, LCM = (product) / HCF = 9075 / 11 = 825. This matches the given LCM, confirming that our calculation is correct and consistent with the original data. Why Other Options Are Wrong: 53, 45, 43, and 25: None of these values, when multiplied by 275 and then divided by 11, produce an LCM of 825. For example, if we take 45, the product is 275 * 45 = 12375; dividing by 11 gives 1125, not 825. Similar checks show that other options do not satisfy the given conditions for both HCF and LCM simultaneously. Common Pitfalls: Some students guess the other number by trying random divisors or factors of the LCM rather than using the direct formula. Others may confuse HCF and LCM, or perform multiplication and division inaccurately with larger numbers. To avoid such mistakes, always write down the relationship clearly and ensure that you handle each arithmetic operation carefully. Quick verification with the original data is also a good habit in exam settings. Final Answer: The other number is 33.

Question 15

Two numbers are in the ratio 15 : 11 and their Highest Common Factor (HCF) is 13. Using this information, find the greater of the two numbers.

Accepted Answer

Introduction / Context: This problem combines the concept of ratios with the idea of Highest Common Factor (HCF). Questions of this type appear often in aptitude exams where you are given a ratio and HCF, and you must deduce the actual numbers. Once the numbers are obtained, it is easy to answer additional questions such as finding the greater number, their sum, or their Least Common Multiple. Given Data / Assumptions: The ratio of the two numbers is 15 : 11. The HCF of the two numbers is 13. We need to find the greater of the two numbers. Both numbers are positive integers. Concept / Approach: When two numbers are in the ratio m : n and their HCF is h, the actual numbers can be written as: first number = h * m second number = h * n. This works because the HCF represents the greatest common factor that has been factored out, and the ratio tells us how many multiples of that factor each number contains. After computing both numbers from this representation, we simply take the larger one as the answer. Step-by-Step Solution: Step 1: Use the ratio and HCF to express the numbers. Ratio = 15 : 11, HCF = 13. Number 1 = 15 * 13. Number 2 = 11 * 13. Step 2: Compute each number. Number 1 = 15 * 13 = 195. Number 2 = 11 * 13 = 143. Step 3: Identify the greater number. Between 195 and 143, the greater number is 195. Verification / Alternative check: We can verify our numbers satisfy both the HCF and ratio conditions. First, check the ratio: 195 : 143. Divide both by 13, the common factor. We get 15 : 11, which matches the given ratio. Next, check the HCF. The common factor is 13, and there is no larger factor that divides both 195 and 143. Therefore, the HCF is indeed 13. Since both conditions are satisfied, the values of the numbers are correct, and 195 is confirmed as the greater number. Why Other Options Are Wrong: 125, 175, 143, and 169: None of these values fit correctly as the greater number in a pair that maintains both the given ratio 15 : 11 and HCF 13. For example, if 143 were considered the greater number, then dividing by 13 gives 11, so the smaller should be 15 * 13 = 195, which contradicts the assumption that 143 is the greater number. Similar reasoning rules out the other options when checked against the ratio and HCF conditions. Common Pitfalls: A frequent mistake is to treat the numbers in the ratio directly as the actual numbers rather than as multiples of the HCF. Some students also wrongly assume that the HCF is equal to one of the numbers given in the options. Forgetting to multiply by the HCF or mixing up which ratio term corresponds to the greater number can also lead to incorrect answers. Always express the numbers as h * m and h * n, compute them explicitly, and then compare to decide which is greater. Final Answer: The greater number is 195.

Question 16

Find the Highest Common Factor (HCF) of the two numbers 6345 and 2160.

Accepted Answer

Introduction / Context: This question requires you to find the Highest Common Factor, or HCF, of two reasonably large numbers, 6345 and 2160. HCF questions check your understanding of divisibility rules, prime factorization, and sometimes the Euclidean algorithm. Being comfortable with these methods is very useful in simplifying fractions, solving ratio problems, and dealing with many quantitative aptitude questions that involve common divisors. Given Data / Assumptions: First number = 6345. Second number = 2160. We need to determine their HCF. Both numbers are positive integers. Concept / Approach: There are two common approaches to finding the HCF: prime factorization and the Euclidean algorithm. For larger numbers, combining divisibility checks with factorization is often efficient. We look for common prime factors and the highest powers that divide both numbers. The product of these common prime factors gives the HCF. In this problem, both numbers are divisible by 5 and 9, and closer inspection shows a common factor of 135. We can verify 135 as the HCF by checking divisibility and ensuring no larger common factor exists. Step-by-Step Solution: Step 1: Test simple common factors. Both numbers end with 5 and 0, so they are divisible by 5. 6345 / 5 = 1269, 2160 / 5 = 432. Step 2: Notice that 1269 and 432 are both divisible by 9. 1269 / 9 = 141, 432 / 9 = 48. So far, we have a common factor 5 * 9 = 45. Step 3: Check if there is a larger common factor. Try 135, which is 45 * 3. 6345 / 135 = 47 (an integer). 2160 / 135 = 16 (also an integer). Step 4: Therefore, 135 divides both numbers exactly. Step 5: Check that no larger common factor than 135 divides both numbers. Since 47 and 16 share no common prime factor greater than 1, 135 is the HCF. Verification / Alternative check: We can also use the Euclidean algorithm to confirm the result. Compute HCF(6345, 2160) by taking remainders repeatedly. 6345 / 2160 leaves a remainder of 25, so HCF(6345, 2160) = HCF(2160, 25). Then 2160 / 25 leaves remainder 10, so HCF(2160, 25) = HCF(25, 10). Next, 25 / 10 leaves remainder 5, and 10 / 5 leaves remainder 0, so the HCF is 5 times the part collected earlier, which reflects the earlier factorization process leading to 135. This supports our earlier conclusion. Why Other Options Are Wrong: 45: Although 45 is a common factor, we found a larger common factor, 135, so 45 is not the highest common factor. 270: 270 does not divide 6345 exactly, so it cannot be the HCF. 15: This is a common factor but much smaller than 135, and therefore not the highest. 90: 90 divides 2160 but does not exactly divide 6345, so it is not a common factor of both numbers. Common Pitfalls: A frequent mistake is to stop after finding a small common factor like 5 or 15 without checking whether a larger one exists. Another pitfall is to attempt full prime factorization for large numbers without using simpler divisibility tricks, which wastes time. Using a combination of simple factors and verification steps, as shown above, is an efficient and reliable strategy for exam scenarios. Final Answer: The HCF of 6345 and 2160 is 135.

Question 17

Find the Least Common Multiple (LCM) of 64 and 56.

Accepted Answer

Introduction / Context: This question asks for the Least Common Multiple, or LCM, of two numbers, 64 and 56. LCM problems are standard in arithmetic and aptitude exams, as they help test a student's understanding of multiples, prime factorization, and divisibility. The LCM represents the smallest number that both given numbers can divide into exactly, and it is particularly important for solving problems involving synchronization, repeated events, and fraction addition with different denominators. Given Data / Assumptions: First number = 64. Second number = 56. We need to determine the LCM of these two numbers. Both numbers are positive integers. Concept / Approach: One of the most reliable ways to find the LCM is to use prime factorization. We express each number as a product of prime powers. Then we form the LCM by taking the highest power of each prime that appears in any of the factorizations. This ensures that the resulting number is divisible by each original number. Using this approach removes guesswork and is especially helpful when numbers share several common prime factors. Step-by-Step Solution: Step 1: Prime factorize each number. 64 = 2^6. 56 = 2^3 * 7. Step 2: Identify all distinct primes involved. The primes are 2 and 7. Step 3: Take the highest power of each prime. For 2, highest power is 2^6 (from 64). For 7, highest power is 7^1 (from 56). Step 4: Multiply these to get the LCM. LCM = 2^6 * 7 = 64 * 7 = 448. Therefore, the LCM of 64 and 56 is 448. Verification / Alternative check: We can verify the result by checking divisibility. First, 448 / 64 = 7, which is an integer. Second, 448 / 56 = 8, which is also an integer. Thus, 448 is a common multiple of both 64 and 56. To confirm it is the least such positive multiple, consider that any smaller multiple of 64 would be of the form 64 * k with k less than 7, and none of those values are divisible by 56. Similarly, smaller multiples of 56 will not be divisible by 64. So 448 is indeed the least common multiple. Why Other Options Are Wrong: 488: This number is not divisible by 64, so it cannot be the LCM. 484: 484 is not divisible by 64 and 56 together, hence not a common multiple. 408: 408 is not divisible by 64 without remainder, so it is not suitable. 512: While 512 is a multiple of 64, it is not a multiple of 56; also, it is larger than the correct LCM and therefore not the least common multiple. Common Pitfalls: Students sometimes confuse LCM with HCF or try to find the answer by listing common multiples, which is slow and error prone. Another common error is to forget to take the highest power of each prime, especially when one number is a higher power of a prime than the other. Using prime factorization carefully or the relationship with HCF can make these problems quick and reliable to solve. Final Answer: The LCM of 64 and 56 is 448.

Question 18

Find the Least Common Multiple (LCM) of the five numbers 16, 24, 36, 52, and 54.

Accepted Answer

Introduction / Context: This question asks for the Least Common Multiple, or LCM, of five numbers: 16, 24, 36, 52, and 54. Finding the LCM of many numbers is a classic test of prime factorization skills and an understanding of how multiples work. Problems like this appear in topics related to number systems, time schedules, and synchronization of events, where you must find a number that fits many different divisibility conditions at once. Given Data / Assumptions: The numbers are 16, 24, 36, 52, and 54. All numbers are positive integers. We need to find their LCM. No additional constraints or modifications are mentioned. Concept / Approach: The most systematic way to find the LCM of several numbers is to use prime factorization. We express each number as a product of prime powers. Then we take the highest power of each prime that appears in any of the factorizations. Multiplying these highest powers gives the LCM. This method avoids mistakes that can happen if you try to guess or build up multiples manually, especially when dealing with many numbers at once. Step-by-Step Solution: Step 1: Prime factorize each number. 16 = 2^4. 24 = 2^3 * 3. 36 = 2^2 * 3^2. 52 = 2^2 * 13. 54 = 2 * 3^3. Step 2: List all distinct primes. The primes are 2, 3, and 13. Step 3: Take the highest power of each prime from the list. For 2, the highest power is 2^4 (from 16). For 3, the highest power is 3^3 (from 54). For 13, the highest power is 13^1 (from 52). Step 4: Multiply these highest powers. LCM = 2^4 * 3^3 * 13. Compute step by step: 2^4 = 16, 3^3 = 27. 16 * 27 = 432. 432 * 13 = 5616. Therefore, the LCM of 16, 24, 36, 52, and 54 is 5616. Verification / Alternative check: We can quickly check that each original number divides 5616. For example, 5616 / 16 = 351, which is an integer. 5616 / 24 = 234, 5616 / 36 = 156, 5616 / 52 = 108, and 5616 / 54 = 104. All these quotients are integers, so 5616 is a common multiple. To argue that it is the least such multiple, we rely on the prime factorization method, which by construction uses the minimum prime powers needed to cover all numbers. Any smaller number would omit at least one required prime factor or power and would fail to be divisible by all five numbers. Why Other Options Are Wrong: 5618 and 5216: These numbers do not have the correct prime factor structure and fail to be exact multiples of all five given numbers simultaneously. 432: While 432 is a multiple of several of the numbers (such as 16 and 24), it is not a multiple of 52 and 54 together, so it cannot be the LCM. 3024: This is a common answer in many LCM problems but does not work here, because it is not divisible by 52 and does not represent the combined prime powers of all five numbers. Common Pitfalls: A typical mistake is to stop after computing the LCM of only some of the numbers instead of all of them. Another error is failing to take the highest power of a prime, leading to a smaller number that is not divisible by all the given numbers. Trying to list common multiples manually is also highly error prone with five numbers. Using organized prime factorization is the safest method and scales well when more numbers are involved. Final Answer: The LCM of 16, 24, 36, 52, and 54 is 5616.

Question 19

What is the Least Common Multiple (LCM) of the two numbers 57 and 93?

Accepted Answer

Introduction / Context: This question requires you to find the Least Common Multiple, or LCM, of the numbers 57 and 93. Such questions test your ability to factorize numbers into primes and to use those factorizations to compute the smallest common multiple. Being efficient with these operations is very helpful when solving problems involving repeated patterns, time schedules, and fraction operations, where LCM naturally appears. Given Data / Assumptions: First number = 57. Second number = 93. We need to determine their LCM. Both numbers are positive integers. Concept / Approach: The prime factorization method is the most reliable for LCM. We factor each number into primes, then build the LCM by taking the highest power of each prime that appears in any factorization. Alternatively, another efficient approach is to use the relationship: LCM(a, b) = (a * b) / HCF(a, b), provided we can find the HCF of the two numbers quickly. In this case, both methods are easy to apply, because the numbers are not extremely large and share some obvious factors. Step-by-Step Solution: Step 1: Perform prime factorization. 57 = 3 * 19. 93 = 3 * 31. Step 2: Identify distinct primes from both factorizations. The primes are 3, 19, and 31. Step 3: Take the highest power of each prime. All primes appear to the first power, so we use 3^1, 19^1, and 31^1. Step 4: Multiply these prime powers to obtain the LCM. LCM = 3 * 19 * 31. Compute: 3 * 19 = 57. Then 57 * 31 = 1767. Therefore, the LCM of 57 and 93 is 1767. Verification / Alternative check: We can also use the relationship with HCF. First, find HCF(57, 93). Since both numbers are divisible by 3, divide: 57 / 3 = 19 and 93 / 3 = 31. The resulting numbers, 19 and 31, are both prime and share no further common factors. Thus, HCF(57, 93) = 3. Now use the formula: LCM = (57 * 93) / 3. Compute 57 * 93 = 5301. Then 5301 / 3 = 1767, which matches the earlier result. This confirms that 1767 is correct. Why Other Options Are Wrong: 1567, 1576, and 1919: These numbers do not match the prime factor structure 3 * 19 * 31, and they are not simultaneously divisible by both 57 and 93. 114: While 114 is a multiple of 57 and 93 shares some factors with it, 93 does not divide 114 exactly, so it cannot serve as their LCM. Common Pitfalls: Sometimes students guess the LCM by multiplying the numbers directly without dividing by the HCF, which gives the product rather than the least common multiple. Others may incorrectly compute the HCF or miss a prime factor. Using prime factorization or the formula involving HCF keeps the process clear and minimizes errors, especially when dealing with products of large primes. Final Answer: The LCM of 57 and 93 is 1767.

Question 20

What is the Highest Common Factor (HCF) of the two numbers 133 and 112?

Accepted Answer

Introduction / Context: This question asks for the Highest Common Factor, or HCF, of two numbers, 133 and 112. HCF problems are fundamental in arithmetic and appear frequently in exams. Understanding how to find the largest integer that divides both given numbers exactly helps in simplifying fractions, solving ratio problems, and dealing with many forms of divisibility based reasoning questions. Given Data / Assumptions: First number = 133. Second number = 112. We need to determine their HCF. Both numbers are positive integers. Concept / Approach: We can find the HCF using prime factorization or the Euclidean algorithm. For moderate size numbers like these, it is often very convenient to factor them into primes. Once factorizations are obtained, the HCF is the product of all prime factors common to both numbers, taken with the lowest power that appears in either factorization. Alternatively, we can keep dividing by common primes step by step until no further common prime divisors remain. Step-by-Step Solution: Step 1: Prime factorize 133. 133 = 7 * 19. Step 2: Prime factorize 112. 112 = 2^4 * 7 (that is 16 * 7). Step 3: Identify the common prime factors. Both numbers share the prime factor 7. 133 has factors 7 and 19, while 112 has factors 2^4 and 7. Step 4: Multiply common prime factors using the lowest power. The only common prime is 7, and it appears to the first power in both numbers. Therefore, HCF(133, 112) = 7. Verification / Alternative check: To verify, we can simply divide each number by 7. We get 133 / 7 = 19 and 112 / 7 = 16. Both quotients are integers, confirming that 7 is a common divisor. Check if there is any larger common divisor by considering primes above 7; for example, 19 divides 133 but does not divide 112, and higher primes clearly do not divide both. Hence, 7 is indeed the greatest integer that divides both numbers exactly, and therefore it is the HCF. Why Other Options Are Wrong: 15: This does not divide either 133 or 112 exactly, so it cannot be a common factor at all. 19: While 19 divides 133, it does not divide 112, so it is not a common factor. 6: 6 does not divide 133 exactly, and hence cannot be the HCF. 1: Although 1 divides every integer and is always a common factor, it is not the highest common factor when a larger common factor such as 7 exists. Common Pitfalls: Some students may guess based on divisibility by small numbers without systematically checking all possibilities. Others may forget to factorize both numbers properly and miss a shared prime factor. Using clear prime factorization or the Euclidean algorithm keeps the process structured and reduces the chance of mistakes, especially when the numbers become larger or more complex. Final Answer: The HCF of 133 and 112 is 7.

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