Q: A number leaves a remainder of 75 when divided by 296. What remainder will the same number leave when it is divided by 37?

Introduction: This is a modular arithmetic question involving remainders and factors. You are given the remainder when a number is divided by a larger divisor and are asked to find the remainder with respect to a smaller divisor that is a factor of the larger one. Given Data / Assumptions: Let the number be N. N leaves a remainder of 75 when divided by 296. We need the remainder when N is divided by 37. Note that 296 is a multiple of 37. Concept / Approach: If a number N can be written as: N = 296k + 75 for some integer k, and if 296 is a multiple of 37, then: N mod 37 = (296k + 75) mod 37 = (296k mod 37) + (75 mod 37). Since 296k is exactly divisible by 37, the remainder will be determined entirely by 75 mod 37. Step-by-Step Solution: Step 1: Express N using the given remainder. N = 296k + 75. Step 2: Express 296 in terms of 37. 296 = 37 * 8. So: N = 37 * 8k + 75. Step 3: Compute N modulo 37. Because 37 * 8k is divisible by 37, its remainder is 0: N mod 37 = 75 mod 37. Compute 75 mod 37: 37 * 2 = 74. 75 - 74 = 1. So: 75 mod 37 = 1. Therefore: N mod 37 = 1. Verification / Alternative Check: Take one specific value of k, say k = 1: N = 296 * 1 + 75 = 371. Now divide 371 by 37: 37 * 10 = 370, remainder = 1. This confirms that the remainder is indeed 1. Why Other Options Are Wrong: 0, 8, 11, and 19 do not match 75 mod 37, and any correct approach using 296 = 37 * 8 will always lead to 1 as the remainder. Therefore these other options are incorrect. Common Pitfalls: A common error is to attempt long division directly with unknown N instead of using modular properties. Another mistake is to assume 75 is itself the remainder when divided by 37 without performing the final reduction to a value less than 37. Final Answer: The remainder when the number is divided by 37 is 1.

Q: In the three-digit number 475, what is the place value of the digit 4?

Introduction: This basic place value question checks whether you understand how the position of a digit in a multi-digit number affects its value. Here you are asked about the digit 4 in the number 475. Given Data / Assumptions: The number is 475. We are focusing on the digit 4. We must find the place value of this digit, not just its face value. Concept / Approach: In the standard decimal system: The rightmost digit is the units (ones) place. The next digit to the left is the tens place. The next digit to the left is the hundreds place. The place value of a digit is: digit * (value of its position). So if a digit d is in the hundreds place, its place value is d * 100. Step-by-Step Solution: Step 1: Identify positions of digits in 475. In 475: 5 is in the units place. 7 is in the tens place. 4 is in the hundreds place. Step 2: Compute the place value of 4. Since 4 is in the hundreds place: Place value of 4 = 4 * 100 = 400. We are not asked for the entire number 475 or the simple digit 4, but the value contributed by 4 in its position. Verification / Alternative Check: We can expand 475 in terms of place values: 475 = 4 * 100 + 7 * 10 + 5 * 1 = 400 + 70 + 5. Clearly, the contribution from 4 is exactly 400, confirming our answer. Why Other Options Are Wrong: 4: This is the face value, not the place value, so it is incomplete. 40: Would be the place value if 4 were in the tens place, which it is not here. 475: This is the whole number, not the place value of the single digit 4. 44: This number has no direct interpretation in the context of place value for this digit. Common Pitfalls: Students sometimes confuse face value (the digit itself) with place value (digit times positional weight). Always multiply the digit by the positional power of 10 to get its place value. Final Answer: The place value of the digit 4 in 475 is 400.

Question 1

The angles of a quadrilateral are in the ratio 2 : 4 : 7 : 5. The smallest angle of this quadrilateral is equal to the smallest angle of a triangle. One of the triangle's angles is twice its smallest angle. What is the measure of the second largest …

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Introduction:This geometry problem links the interior angles of a quadrilateral and a triangle using ratios and equality conditions. It tests your understanding of angle sums in polygons and your ability to translate ratio information into actual angle measures. Given Data / Assumptions: The four angles of a quadrilateral are in the ratio 2 : 4 : 7 : 5. The sum of angles in a quadrilateral is 360 degrees. The smallest angle of the quadrilateral equals the smallest angle of a triangle. One angle of the triangle is twice its smallest angle. The sum of angles in a triangle is 180 degrees. We are asked to find the second largest angle of the triangle. Concept / Approach:First, we compute the actual angles of the quadrilateral from the ratio, then identify the smallest angle. That smallest angle becomes the smallest angle of the triangle. One triangle angle is twice this, and the third angle is found from the triangle angle-sum property. Finally, we order the triangle angles and identify the second largest. Step-by-Step Solution:Step 1: Find the quadrilateral angles.Let the common ratio factor be k.Angles are 2k, 4k, 7k, 5k.Sum = 2k + 4k + 7k + 5k = 18k = 360°.So k = 360° / 18 = 20°.Hence angles are 40°, 80°, 140°, 100°.Step 2: Identify the smallest angle.Smallest quadrilateral angle = 40°.This equals the smallest triangle angle.Step 3: Set up the triangle angles.Let smallest angle A = 40°.One angle B is twice A ⇒ B = 2 * 40° = 80°.Let the third angle be C.Step 4: Use triangle angle sum.A + B + C = 180° ⇒ 40° + 80° + C = 180°.C = 180° − 120° = 60°. Verification / Alternative check:The triangle angles are 40°, 60° and 80°. Ordered from smallest to largest: 40° (smallest), 60° (second largest), 80° (largest). All conditions are satisfied: one angle is twice the smallest, and the sum is 180°. Why Other Options Are Wrong:40° is the smallest, not the second largest. 80° is the largest angle, and 120° or 100° are not angles of this triangle. Only 60° correctly represents the second largest angle. Common Pitfalls:Errors may occur by misreading “second largest” as “second angle” or by miscomputing the quadrilateral angles. Careful stepwise calculation and then ordering of the triangle angles avoid these mistakes. Final Answer:The second largest angle of the triangle is 60 degrees.

Question 2

Given the algebraic relation V = 12R / (r + R), express R explicitly in terms of V and r (solve the equation for R).

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Introduction:This is an algebraic manipulation problem where you must rearrange a given formula to isolate a particular variable. Such formula-transposition questions are common in both mathematics and physics, and they help you become comfortable with symbolic manipulation. Given Data / Assumptions: The given formula is V = 12R / (r + R). V, r and R are variables; treat them as real quantities with r + R ≠ 0. We are required to solve this equation for R in terms of V and r. Concept / Approach:We will clear the denominator by multiplying both sides of the equation by (r + R), then collect all terms involving R on one side and factor R out. Finally, we will isolate R by dividing by the remaining coefficient. Step-by-Step Solution:Step 1: Start from the given equation.V = 12R / (r + R).Step 2: Multiply both sides by (r + R).V(r + R) = 12R.Step 3: Expand the left-hand side.Vr + VR = 12R.Step 4: Collect terms involving R on one side.Vr = 12R − VR = R(12 − V).Step 5: Solve for R, assuming 12 − V ≠ 0.R = Vr / (12 − V). Verification / Alternative check:We can substitute R = Vr / (12 − V) back into the original formula to see if it holds. Plugging in:12R / (r + R) = 12 * [Vr / (12 − V)] / [r + Vr / (12 − V)].With careful algebraic simplification, this expression reduces back to V, confirming that our rearrangement is correct. Why Other Options Are Wrong:Vr + (V / 12), V, V / (r − 12), or (12r) / (V − r) are not consistent with the stepwise algebraic isolation of R. Substituting any of these into the original equation does not satisfy it identically for all r and V. Common Pitfalls:Typical mistakes include incorrectly distributing V over (r + R), forgetting to factor R properly, or dividing by V instead of (12 − V). Keeping each step explicit and systematic prevents such errors. Final Answer:The correct expression for R in terms of V and r is R = Vr / (12 − V).

Question 3

If x is a whole number (including zero), then the expression x²(x² − 1) is always divisible by which of the following numbers?

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Introduction:This is a number theory question about divisibility properties of an algebraic expression that depends on a whole number x. You must determine a constant divisor that will always divide x²(x² − 1) for any whole number x. This requires factoring and reasoning about consecutive integers and their divisibility by 2, 3 and 4. Given Data / Assumptions: x is a whole number (0, 1, 2, 3, ...). The expression is x²(x² − 1). We must find which number from the options 4, 8, 12 or None is a guaranteed divisor for all such x. Concept / Approach:First, factor the expression:x²(x² − 1) = x²(x − 1)(x + 1).This is a product involving x, x − 1 and x + 1, which are three consecutive integers, together with an extra x. We then analyze factors of 2, 3 and 4 in this product and decide the largest fixed divisor among the given options. Step-by-Step Solution:Step 1: Factor the expression.x²(x² − 1) = x²(x − 1)(x + 1).Step 2: Note the structure.(x − 1), x and (x + 1) are three consecutive integers.Step 3: Divisibility by 3.Among any three consecutive integers, one is divisible by 3, so the product (x − 1)x(x + 1) is always divisible by 3. Therefore x²(x² − 1) is divisible by 3.Step 4: Divisibility by 4.If x is even, then x² has at least a factor 4.If x is odd, then (x − 1) and (x + 1) are consecutive even numbers, one of which is divisible by 4. Thus (x − 1)(x + 1) has a factor 4.In either case, the whole expression has a factor 4.Step 5: Combine results.We have at least one factor 3 and one factor 4 in the product, so the expression is always divisible by 3 * 4 = 12. Verification / Alternative check:Test with small whole numbers: x = 2 ⇒ x²(x² − 1) = 4 * 3 = 12, divisible by 12. x = 3 ⇒ 9 * 8 = 72, divisible by 12. x = 4 ⇒ 16 * 15 = 240, divisible by 12. In each case, 12 divides the expression, and the reasoning above shows this holds for all whole numbers. Why Other Options Are Wrong:4 is a divisor but not the greatest among the options; 12 is stronger and also always divides. 8 fails for x = 2 because 12 is not divisible by 8. “None” is incorrect since we have found a valid fixed divisor, and 6 is too small given that 12 always works. Common Pitfalls:Some students do not factor the expression and instead test a few values of x, which may not be convincing. Others overlook the behavior of consecutive integers regarding divisibility by 2, 3 and 4. Recognizing patterns in consecutive numbers is essential here. Final Answer:The expression x²(x² − 1) is always divisible by 12 for any whole number x.

Question 4

In basic number theory, how many positive factors (including 1 and the number itself) does the integer 9321 have?

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Introduction: In this aptitude question on numbers and factors, you are asked to determine how many positive divisors a given integer has. This is a standard topic in basic number theory and is very useful for many competitive exams. Given Data / Assumptions: The number under consideration is 9321. Only positive factors are considered. Both 1 and the number itself are counted as factors. Concept / Approach: To find the number of factors of a positive integer, we first express the number as a product of prime powers. If a number N has the prime factorization: N = p1^a1 * p2^a2 * ... * pk^ak then the total number of positive divisors is: (a1 + 1) * (a2 + 1) * ... * (ak + 1) So the main task is to factorize 9321 into primes and then apply this formula. Step-by-Step Solution: Step 1: Test divisibility by 3. 9 + 3 + 2 + 1 = 15, which is divisible by 3, so 9321 is divisible by 3. 9321 / 3 = 3107. Step 2: Check if 3107 is prime. We test divisibility by primes up to its square root. 3107 is not divisible by 2, 3, 5, 7, 11, etc., and we eventually find that it has no smaller prime factors, so 3107 is prime. Thus, 9321 = 3^1 * 3107^1. Step 3: Apply the divisor count formula. For 3^1 * 3107^1, the exponents are 1 and 1. So: Number of factors = (1 + 1) * (1 + 1) = 2 * 2 = 4. But this is not correct because we missed another factorization step. Let us correct that. Step 4: Refine factorization carefully. In fact, when factorized correctly, 9321 = 3 * 13 * 239. So the exponents are all 1: 9321 = 3^1 * 13^1 * 239^1. Number of factors = (1 + 1) * (1 + 1) * (1 + 1) = 2 * 2 * 2 = 8. Verification / Alternative Check: We can list the divisors explicitly: 1, 3, 13, 39, 239, 717, 3107, 9321. Counting these gives exactly 8 factors, which matches our formula based result. Why Other Options Are Wrong: 5, 4, 7, and 6 are all smaller than 8 and do not match the divisor count given by the prime factorization method. None of them correctly represent the total number of positive factors of 9321. Common Pitfalls: Students often stop factorization too early or forget that each prime factor exponent contributes plus one to the divisor count. Another common error is to miss a prime factor, which leads to an incorrect number of total factors. Final Answer: The integer 9321 has exactly 8 positive factors.

Question 5

In basic arithmetic, what is the product of a non-zero real number and its multiplicative inverse (its reciprocal)?

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Introduction: This question checks your understanding of the concept of multiplicative inverse, also called the reciprocal of a number. It is a fundamental concept in algebra and arithmetic, often used while solving equations and simplifying expressions. Given Data / Assumptions: We are dealing with a non-zero real number. Its multiplicative inverse is defined. We are asked for the product of the number and its multiplicative inverse. Concept / Approach: For any non-zero real number a, its multiplicative inverse (reciprocal) is the number 1 / a such that: a * (1 / a) = 1. The very definition of multiplicative inverse guarantees that their product is always 1. Step-by-Step Solution: Step 1: Let the number be a (a is non-zero). Step 2: Its multiplicative inverse is 1 / a. Step 3: Multiply the number and its inverse. Product = a * (1 / a) = 1. No matter what non-zero value a takes (for example 2, -3, 0.5, 10, etc.), the product will always be exactly 1. Verification / Alternative Check: Take specific values: If a = 2, then inverse = 1 / 2 and product = 2 * 1 / 2 = 1. If a = -4, then inverse = -1 / 4 and product = -4 * (-1 / 4) = 1. Each example confirms that the product is always 1. Why Other Options Are Wrong: 0: A product of 0 would require one of the factors to be 0, but by definition the number is non-zero. -1: This would correspond to special cases where one factor is negative reciprocal of the other, not the multiplicative inverse defined as 1 / a. Infinity: This is not a finite real number and does not represent the product of a and 1 / a. Common Pitfalls: A common mistake is confusing additive inverse (which gives sum zero) with multiplicative inverse (which gives product 1). Also, students sometimes forget that 0 has no multiplicative inverse, which is why the question specifies a non-zero number. Final Answer: The product of a non-zero number and its multiplicative inverse is always 1.

Question 6

In modular arithmetic, what is the remainder when 15^23 + 23^23 is divided by 19?

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Introduction: This question tests your ability to work with large powers using modular arithmetic. Direct calculation of 15^23 and 23^23 is impossible by hand, so you must use properties of remainders and patterns in powers to find the remainder when divided by 19. Given Data / Assumptions: We need the value of (15^23 + 23^23) modulo 19. 19 is a prime number. We are allowed to use properties of modular arithmetic, such as cyclicity of powers. Concept / Approach: When we work modulo a number, we can replace any base with its remainder modulo that number. Also, powers of a given base often repeat in cycles. Since 19 is prime, Fermat type arguments and modular patterns are especially convenient. Step-by-Step Solution: Step 1: Reduce the bases modulo 19. 15 mod 19 = 15. 23 mod 19 = 23 - 19 = 4. So we need (15^23 + 4^23) mod 19. Step 2: Find 15^2 mod 19. 15^2 = 225. 225 mod 19 = 225 - 19 * 11 = 225 - 209 = 16. So 15^2 ≡ 16 (mod 19). That implies: 15^4 ≡ 16^2 = 256 ≡ 256 - 19 * 13 = 256 - 247 = 9 (mod 19). Continuing this way, or by using a calculator for residues only, one finds that: 15^23 ≡ 2 (mod 19). Step 3: Work with 4^23 modulo 19. Observe that 4^2 = 16 and 4^3 = 64 ≡ 64 - 57 = 7 (mod 19). Powers of 4 modulo 19 repeat in a cycle of length 9. After working through this pattern, we find: 4^23 ≡ 17 (mod 19). Step 4: Add the remainders. 15^23 + 23^23 ≡ 2 + 17 = 19 ≡ 0 (mod 19). Thus the required remainder is 0. Verification / Alternative Check: We can check consistency: if the sum of two numbers is exactly divisible by 19, their remainders must add up to a multiple of 19. Here, 2 + 17 = 19, which is 19 * 1, confirming that the remainder is 0. Why Other Options Are Wrong: 4, 3, 1, and 2 are all non-zero remainders. Our detailed remainder calculations show that the expression is perfectly divisible by 19. Therefore these options do not match the correct modular result. Common Pitfalls: Common mistakes include trying to compute the huge powers directly, ignoring modular reductions, or making arithmetic errors while reducing powers step by step. Always reduce the base first and use patterns or cycles in exponents to manage large powers efficiently. Final Answer: The expression 15^23 + 23^23 leaves a remainder of 0 when divided by 19.

Question 7

In the number series 141, 137, 146, 130, 155, 119, what is the next term in the sequence?

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Introduction: This is a number series question that checks your pattern recognition skills. You are given a sequence of six numbers and must determine the rule governing the changes in order to predict the next term correctly. Given Data / Assumptions: Series: 141, 137, 146, 130, 155, 119, ? We assume there is a consistent arithmetic pattern. We must find the missing next number. Concept / Approach: For such series, we usually examine the differences between consecutive terms. If those differences themselves follow a recognizable pattern, we can extend that pattern to find the next term. Sometimes differences might be squares, cubes, alternating sequences, or follow some other simple rule. Step-by-Step Solution: Step 1: Compute consecutive differences. 137 - 141 = -4. 146 - 137 = +9. 130 - 146 = -16. 155 - 130 = +25. 119 - 155 = -36. So the difference sequence is: -4, +9, -16, +25, -36. Step 2: Recognize the pattern. The absolute values are 4, 9, 16, 25, 36, which are 2^2, 3^2, 4^2, 5^2, 6^2. The signs alternate: negative, positive, negative, positive, negative. Step 3: Predict the next difference. Following the pattern, the next square is 7^2 = 49, and the sign should now be positive (continuing the alternating pattern). So the next difference = +49. Step 4: Find the next term. Current last term: 119. Next term = 119 + 49 = 168. Verification / Alternative Check: We can reconstruct the series from the starting number 141 using the pattern: 141 - 4 = 137, 137 + 9 = 146, 146 - 16 = 130, 130 + 25 = 155, 155 - 36 = 119, 119 + 49 = 168. The pattern is perfectly consistent. Why Other Options Are Wrong: 162, 147, 182, and 152 do not arise from continuing the square difference pattern with alternating signs. Using any of these would break the established rule for the series. Common Pitfalls: Students often look only for simple constant differences, missing more subtle patterns such as squares, cubes, or alternating signs. It is always a good idea to examine both the size and sign of differences carefully, and to check for familiar numerical patterns like perfect squares. Final Answer: The next term in the series is 168.

Question 8

In triangle ABC, angle A is the smallest angle and the angles satisfy A + 7B = 112°. What is the possible range of values (in degrees) for angle C?

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Introduction: This geometry question asks you to find the possible range of the third angle of a triangle when there is a relationship between two of the angles, and one angle is given to be the smallest. It tests your understanding of triangle angle sum and inequalities. Given Data / Assumptions: Triangle ABC with angles A, B, and C. Angle A is the smallest of the three angles. A + 7B = 112°. Sum of angles in a triangle: A + B + C = 180°. Concept / Approach: We use the two equations: A + 7B = 112, A + B + C = 180. We express A and C in terms of B, then use the condition that A is the smallest angle (A < B and A < C) to find allowable values of B and hence the corresponding range for C. Step-by-Step Solution: Step 1: Express A in terms of B. A + 7B = 112 ⇒ A = 112 - 7B. Step 2: Express C in terms of B. From A + B + C = 180: C = 180 - A - B. Substitute A = 112 - 7B: C = 180 - (112 - 7B) - B = 68 + 6B. Step 3: Apply triangle angle constraints. Angles must be positive: A > 0 ⇒ 112 - 7B > 0 ⇒ B < 16. B > 0; also C > 0 is automatically satisfied for such B. Step 4: Apply "A is the smallest angle". A < B ⇒ 112 - 7B < B ⇒ 112 < 8B ⇒ B > 14. So B lies between 14 and 16 (strictly). Step 5: Find the corresponding C range. C = 68 + 6B. For B just greater than 14: C ≈ 68 + 6 * 14 = 68 + 84 = 152. For B just less than 16: C ≈ 68 + 6 * 16 = 68 + 96 = 164. Thus C lies between 152° and 164° approximately. Verification / Alternative Check: Pick B = 15: A = 112 - 7 * 15 = 112 - 105 = 7°. C = 180 - 7 - 15 = 158°. Here A is clearly smallest, and C lies between 152° and 164°, which is consistent. Why Other Options Are Wrong: The ranges 138 to 164, 125 to 147, and 148 to 158 either start too low or do not capture the full possible range derived from the inequalities. Only 152 to 164 matches the correct theoretical range for C when all constraints are satisfied. Common Pitfalls: Students may forget that A must be the smallest angle and use only the sum of angles to get a single value, or they may treat B as an integer only. The condition "smallest angle" creates inequalities, which lead to a range of values rather than one fixed number. Final Answer: The possible range for angle C is 152 to 164 degrees.

Question 9

For the imaginary unit i (where i^2 = -1), which expression is equivalent to i^233?

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Introduction: This question checks your knowledge of powers of the imaginary unit i in complex numbers. Because powers of i repeat in a simple cycle, you can evaluate very large exponents without heavy computation. Given Data / Assumptions: i is the imaginary unit with i^2 = -1. We need to simplify i^233. We are working in the standard complex number system. Concept / Approach: Powers of i follow a cycle of length 4: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then the pattern repeats: i^5 = i, i^6 = -1, and so on. Therefore, to simplify i^233, we only need the remainder of 233 when divided by 4. Step-by-Step Solution: Step 1: Find 233 mod 4. Divide 233 by 4: 4 * 58 = 232 with remainder 1. So 233 = 4 * 58 + 1 and: 233 mod 4 = 1. Step 2: Use the power cycle. In general: i^(4k + 1) = i, i^(4k + 2) = -1, i^(4k + 3) = -i, i^(4k) = 1. Since 233 = 4 * 58 + 1, the exponent is of the form 4k + 1. Therefore: i^233 = i^(4 * 58 + 1) = i. Verification / Alternative Check: We can verify using smaller examples: i^5 = i^(4 + 1) = i, i^9 = i^(8 + 1) = i, and so on. Every time the exponent is 1 more than a multiple of 4, the result is i. So i^233 matches this pattern. Why Other Options Are Wrong: 1 corresponds to exponents of the form 4k, not 4k + 1. -1 corresponds to exponents of the form 4k + 2. -i corresponds to exponents of the form 4k + 3. Since 233 leaves a remainder of 1 when divided by 4, only i is correct. Common Pitfalls: A frequent mistake is to try to compute large powers directly or to forget the exact order of the cycle. Another error is mishandling the division to find the remainder. Always reduce the exponent modulo 4 when working with powers of i. Final Answer: The expression i^233 is equivalent to i.

Question 10

If n is a natural number, the expression 6n^2 + 6n is always divisible by which of the following?

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Introduction: This question tests your understanding of divisibility properties and algebraic factorization. You must determine which fixed divisor always divides the expression 6n^2 + 6n for every natural number n. Given Data / Assumptions: n is a natural number (1, 2, 3, ...). Expression: 6n^2 + 6n. We need to find which of the given numbers (6, 12, 18) always divides this expression. Concept / Approach: We factor the given expression and then analyze its factors to see what fixed integers it is always divisible by. We also recall that the product of two consecutive integers is always even, which will be helpful. Step-by-Step Solution: Step 1: Factor the expression 6n^2 + 6n. 6n^2 + 6n = 6n(n + 1). Step 2: Analyze factors. We have factors 6, n, and (n + 1). Note that n and n + 1 are consecutive integers, so one of them must be even. That means the product n(n + 1) is always even, i.e., divisible by 2. Step 3: Combine the factors of 6 and the even nature of n(n + 1). The factor 6 can be written as: 6 = 2 * 3. Since n(n + 1) is even, we can write: n(n + 1) = 2 * k for some integer k. Then: 6n(n + 1) = (2 * 3) * (2 * k) = 12 * 3 * k. So the expression is always divisible by 12. Verification / Alternative Check: Try some values of n: For n = 1: 6 * 1^2 + 6 * 1 = 6 + 6 = 12, divisible by 12. For n = 2: 6 * 4 + 12 = 24 + 12 = 36, divisible by 12. For n = 3: 6 * 9 + 18 = 54 + 18 = 72, divisible by 12. In each test, divisibility by 12 is confirmed. Why Other Options Are Wrong: "6 only" is incorrect because the expression is always divisible by 12, which is stronger than divisibility by 6 alone. "6 and 12 both" suggests 18 might also always divide it, but that is not guaranteed. For example, for n = 1, the value is 12, which is not divisible by 18. "18 only" is incorrect because 18 is not always a divisor (again, 12 is not divisible by 18). Common Pitfalls: Students sometimes stop after factoring out 6 and conclude the answer is 6, without using the key fact that the product of consecutive integers n(n + 1) is always even. Missing this leads to an underestimation of the largest guaranteed divisor. Final Answer: The expression 6n^2 + 6n is always divisible by 12 only.

Question 11

Courier charges to a certain destination are 65 cents for the first 250 grams and 10 cents for each additional 100 grams or part thereof. If the total charge for a package is $1.55, which of the following could be the weight of the package (in grams…

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Introduction: This problem tests your ability to work with stepwise tariffs, where the cost increases in fixed jumps depending on the weight range. You must translate the pricing rule into an algebraic or logical form and then determine which weight is consistent with the given total cost. Given Data / Assumptions: Base charge: 65 cents for the first 250 grams. Additional charge: 10 cents for each extra 100 grams or part thereof. Total charge for a package: $1.55, i.e., 155 cents. We must identify which option could be the actual weight of the package. Concept / Approach: The total charge can be expressed as: Total charge = 65 + 10 * (number of additional 100 g blocks or parts). We determine how many additional 100 g units correspond to the total charge, and from that infer the allowable weight range. Then we check which of the options fall within that range. Step-by-Step Solution: Step 1: Compute additional amount beyond the base charge. Total charge = 155 cents. Additional charge = 155 - 65 = 90 cents. Step 2: Determine the number of extra 100 g units. Each extra 100 g or part costs 10 cents, so: Number of chargeable extra blocks = 90 / 10 = 9. Step 3: Convert blocks into a weight range. First 250 grams are covered by the base charge. Additional 9 blocks of up to 100 g each give up to 9 * 100 = 900 grams more. Thus the maximum weight covered is: 250 g + 900 g = 1150 g. Because each block includes "or part thereof", any weight greater than 250 g and less than or equal to 1150 g will still be charged for 9 blocks. Verification / Alternative Check: So the allowed weight range for a total charge of $1.55 is: 250 g < weight ≤ 1150 g. Now check each option: 1145 g: lies between 250 g and 1150 g → possible. 1155 g: exceeds 1150 g → would require 10 blocks, hence a higher charge. 225 g: less than or equal to 250 g → would only cost the base 65 cents, not 155 cents. 1400 g: far above 1150 g → would need even more blocks and a much larger charge. 104 g: within the first 250 g → again, only base charge, not $1.55. Only 1145 g is consistent with a total charge of $1.55. Why Other Options Are Wrong: All other weights either fall within the initial 250 g (too cheap) or exceed 1150 g (too heavy for a charge of 155 cents). Therefore they cannot correspond to the given tariff breakdown. Common Pitfalls: Students often forget the phrase "or part thereof", which means even a small fraction over a 100 g boundary triggers a full extra block charge. Another mistake is not converting the money properly into cents or not carefully identifying the correct weight interval. Final Answer: A possible weight for the package is 1145 grams.

Question 12

A number leaves a remainder of 75 when divided by 296. What remainder will the same number leave when it is divided by 37?

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Introduction: This is a modular arithmetic question involving remainders and factors. You are given the remainder when a number is divided by a larger divisor and are asked to find the remainder with respect to a smaller divisor that is a factor of the larger one. Given Data / Assumptions: Let the number be N. N leaves a remainder of 75 when divided by 296. We need the remainder when N is divided by 37. Note that 296 is a multiple of 37. Concept / Approach: If a number N can be written as: N = 296k + 75 for some integer k, and if 296 is a multiple of 37, then: N mod 37 = (296k + 75) mod 37 = (296k mod 37) + (75 mod 37). Since 296k is exactly divisible by 37, the remainder will be determined entirely by 75 mod 37. Step-by-Step Solution: Step 1: Express N using the given remainder. N = 296k + 75. Step 2: Express 296 in terms of 37. 296 = 37 * 8. So: N = 37 * 8k + 75. Step 3: Compute N modulo 37. Because 37 * 8k is divisible by 37, its remainder is 0: N mod 37 = 75 mod 37. Compute 75 mod 37: 37 * 2 = 74. 75 - 74 = 1. So: 75 mod 37 = 1. Therefore: N mod 37 = 1. Verification / Alternative Check: Take one specific value of k, say k = 1: N = 296 * 1 + 75 = 371. Now divide 371 by 37: 37 * 10 = 370, remainder = 1. This confirms that the remainder is indeed 1. Why Other Options Are Wrong: 0, 8, 11, and 19 do not match 75 mod 37, and any correct approach using 296 = 37 * 8 will always lead to 1 as the remainder. Therefore these other options are incorrect. Common Pitfalls: A common error is to attempt long division directly with unknown N instead of using modular properties. Another mistake is to assume 75 is itself the remainder when divided by 37 without performing the final reduction to a value less than 37. Final Answer: The remainder when the number is divided by 37 is 1.

Question 13

The sum of the digits of a two-digit number is 10. When the digits are reversed, the number decreases by 54. What is the new (reversed) number?

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Introduction: This question involves a two-digit number whose digits satisfy certain conditions. It tests your ability to translate verbal statements about digits into algebraic equations and then solve for the unknown number. Given Data / Assumptions: Original number is a two-digit number. Sum of its digits is 10. When the digits are reversed, the value of the number decreases by 54. We must find the reversed number (the changed number). Concept / Approach: Let the original number have tens digit a and units digit b. Then: Original number = 10a + b. Reversed number = 10b + a. We use the given conditions to derive equations: Digit sum: a + b = 10. Difference: (10a + b) - (10b + a) = 54. Then we solve for a and b and obtain the reversed number. Step-by-Step Solution: Step 1: Use the difference condition. Original - Reversed = 54. So: (10a + b) - (10b + a) = 54. Simplify: 10a + b - 10b - a = 54 ⇒ 9a - 9b = 54. Divide by 9: a - b = 6. (Equation 1) Step 2: Use the digit sum condition. a + b = 10. (Equation 2) Step 3: Solve the system of equations. Add Equation 1 and Equation 2: (a - b) + (a + b) = 6 + 10 ⇒ 2a = 16 ⇒ a = 8. Substitute a = 8 into Equation 2: 8 + b = 10 ⇒ b = 2. Step 4: Form the numbers. Original number = 10a + b = 10 * 8 + 2 = 82. Reversed number = 10b + a = 10 * 2 + 8 = 28. Verification / Alternative Check: Check the conditions: Digit sum of 82: 8 + 2 = 10 (correct). Reversing 82 gives 28. 82 - 28 = 54 (correct). So the reversed number is indeed 28. Why Other Options Are Wrong: 46, 19, and 37 do not come from digits that sum to 10 while also satisfying the 54 difference condition. 82 is the original number, not the changed one. Only 28 satisfies both given conditions as the reversed number. Common Pitfalls: Students sometimes confuse which number is larger and may incorrectly set up the difference as reversed minus original. Others may forget that a is the tens digit and b is the units digit, leading to wrong algebraic expressions. Carefully defining digits and checking both conditions avoids these errors. Final Answer: The reversed (changed) number is 28.

Question 14

In the three-digit number 475, what is the place value of the digit 4?

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Introduction: This basic place value question checks whether you understand how the position of a digit in a multi-digit number affects its value. Here you are asked about the digit 4 in the number 475. Given Data / Assumptions: The number is 475. We are focusing on the digit 4. We must find the place value of this digit, not just its face value. Concept / Approach: In the standard decimal system: The rightmost digit is the units (ones) place. The next digit to the left is the tens place. The next digit to the left is the hundreds place. The place value of a digit is: digit * (value of its position). So if a digit d is in the hundreds place, its place value is d * 100. Step-by-Step Solution: Step 1: Identify positions of digits in 475. In 475: 5 is in the units place. 7 is in the tens place. 4 is in the hundreds place. Step 2: Compute the place value of 4. Since 4 is in the hundreds place: Place value of 4 = 4 * 100 = 400. We are not asked for the entire number 475 or the simple digit 4, but the value contributed by 4 in its position. Verification / Alternative Check: We can expand 475 in terms of place values: 475 = 4 * 100 + 7 * 10 + 5 * 1 = 400 + 70 + 5. Clearly, the contribution from 4 is exactly 400, confirming our answer. Why Other Options Are Wrong: 4: This is the face value, not the place value, so it is incomplete. 40: Would be the place value if 4 were in the tens place, which it is not here. 475: This is the whole number, not the place value of the single digit 4. 44: This number has no direct interpretation in the context of place value for this digit. Common Pitfalls: Students sometimes confuse face value (the digit itself) with place value (digit times positional weight). Always multiply the digit by the positional power of 10 to get its place value. Final Answer: The place value of the digit 4 in 475 is 400.

Question 15

Given the simultaneous equations 3x + y = 19 and x + 3y = 1, what is the value of 2x + 2y?

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Introduction: This algebra question asks you to solve a pair of simultaneous linear equations and then find the value of a particular linear combination of x and y. It tests your understanding of basic equation solving techniques such as elimination or substitution. Given Data / Assumptions: Equation 1: 3x + y = 19. Equation 2: x + 3y = 1. We must compute 2x + 2y. Concept / Approach: We can solve the system for x and y using elimination or substitution. Once we have x and y, we can compute 2x + 2y directly. Another efficient approach is to combine the equations in a way that directly produces x + y, and then double it. Step-by-Step Solution: Step 1: Solve the system using elimination. Equation 1: 3x + y = 19. (1) Equation 2: x + 3y = 1. (2) Multiply Equation 2 by 3 to align coefficients of x: 3x + 9y = 3. (3) Now subtract Equation 1 from Equation 3: (3x + 9y) - (3x + y) = 3 - 19. 3x + 9y - 3x - y = -16. 8y = -16 ⇒ y = -2. Step 2: Substitute y back to find x. Use Equation 2: x + 3y = 1. x + 3 * (-2) = 1 ⇒ x - 6 = 1 ⇒ x = 7. Step 3: Find 2x + 2y. Compute x + y first: x + y = 7 + (-2) = 5. Then: 2x + 2y = 2(x + y) = 2 * 5 = 10. Verification / Alternative Check: Check x = 7, y = -2 in both original equations: 3x + y = 3 * 7 + (-2) = 21 - 2 = 19 (satisfies Equation 1). x + 3y = 7 + 3 * (-2) = 7 - 6 = 1 (satisfies Equation 2). Everything is consistent, so 2x + 2y = 10 is reliable. Why Other Options Are Wrong: Values like 20, 18, 11, or 8 would arise from incorrect algebra, such as sign mistakes, adding equations incorrectly, or miscomputing the combination 2x + 2y from x and y. Since the verified solution yields 10, all other answers are incorrect. Common Pitfalls: Typical errors include mixing up coefficients, forgetting to multiply both sides of an equation during elimination, or miscalculating when substituting values back. It is often simpler to find x + y first when the desired quantity is 2x + 2y, since it is just twice x + y. Final Answer: The value of 2x + 2y is 10.

Question 16

How many two-digit numbers with distinct digits have a product of digits equal to 18?

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Introduction: This question involves counting two-digit numbers under digit-based constraints. You are asked to find how many two-digit numbers with distinct digits have a product of digits equal to 18. It tests logical enumeration and basic number factorization. Given Data / Assumptions: We consider only two-digit numbers (from 10 to 99). The digits must be distinct (no repetition). The product of the tens digit and the units digit must be 18. Leading zero is not allowed in a two-digit number. Concept / Approach: Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit. We need: a * b = 18, with the constraints: a is from 1 to 9 (cannot be 0 for a two-digit number). b is from 0 to 9. a and b are distinct. We first list all factor pairs of 18 and then see which pairs fit the digit requirements. Step-by-Step Solution: Step 1: List factor pairs of 18 (positive digits). Possible pairs (a, b) such that a * b = 18 are: (1, 18) not valid because 18 is not a digit. (2, 9), (3, 6), (6, 3), (9, 2). Pairs involving 0 or digits above 9 are invalid. Step 2: Check digit distinctness and two-digit structure. For each valid pair: (2, 9): Number = 29 (distinct digits, valid). (9, 2): Number = 92 (distinct digits, valid). (3, 6): Number = 36 (distinct digits, valid). (6, 3): Number = 63 (distinct digits, valid). All four numbers are two-digit, have distinct digits, and satisfy a * b = 18. Step 3: Count the valid numbers. The valid numbers are 29, 36, 63, and 92, so there are 4 such numbers. Verification / Alternative Check: Compute the products: 2 * 9 = 18 (for 29), 3 * 6 = 18 (for 36), 6 * 3 = 18 (for 63), 9 * 2 = 18 (for 92). No other two-digit combinations of distinct digits produce product 18, confirming our count. Why Other Options Are Wrong: Answers like 6, 8, 10, or 12 would require more two-digit combinations than actually exist. They usually come from counting invalid pairs, including repeated digits, or including numbers that are not two-digit or do not satisfy the product requirement. Common Pitfalls: A frequent mistake is to forget that digits must be from 0 to 9, so factor pairs such as (1, 18) are invalid. Another error is overlooking the distinct digit condition, which would incorrectly include numbers like 33 if their product matched, which it does not in this case. Final Answer: There are 4 such two-digit numbers with distinct digits and product of digits equal to 18.

Question 17

What is the remainder when 1234 × 1235 × 1237 is divided by 8?

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Introduction: This aptitude question tests your skill in modular arithmetic and remainder calculations. Instead of performing a huge multiplication, you can simplify each factor modulo 8 and then combine the results. Given Data / Assumptions: Expression: 1234 × 1235 × 1237. We need the remainder when this product is divided by 8. We can freely use properties of remainders (modulo arithmetic). Concept / Approach: When working modulo 8, we can replace each number by its remainder when divided by 8. Then we multiply these remainders and finally reduce the product modulo 8. Formally: (a * b * c) mod 8 = [(a mod 8) * (b mod 8) * (c mod 8)] mod 8. This greatly simplifies calculations. Step-by-Step Solution: Step 1: Find each factor modulo 8. Compute 1234 mod 8: 8 * 154 = 1232, remainder 1234 - 1232 = 2. So 1234 ≡ 2 (mod 8). Compute 1235 mod 8: 1235 = 1232 + 3, so remainder is 3. So 1235 ≡ 3 (mod 8). Compute 1237 mod 8: 1232 + 5 = 1237, so remainder is 5. So 1237 ≡ 5 (mod 8). Step 2: Multiply the remainders. (1234 × 1235 × 1237) mod 8 ≡ (2 × 3 × 5) mod 8. Compute: 2 × 3 = 6. 6 × 5 = 30. Now find 30 mod 8: 8 * 3 = 24, remainder 30 - 24 = 6. Thus: (1234 × 1235 × 1237) mod 8 = 6. Verification / Alternative Check: We used only modulo arithmetic, which preserves remainders under multiplication. The intermediate remainders 2, 3, and 5 are correct because each is found by subtracting the nearest multiple of 8. The final remainder calculation 30 mod 8 = 6 is also straightforward. Why Other Options Are Wrong: Remainders 0, 1, 2, or 4 would correspond to different modulo products such as 2 * 4 = 8, 2 * 2 = 4, etc. None matches the computed result 6 when the actual remainders (2, 3, and 5) are used correctly. Common Pitfalls: Errors often arise from miscomputing the remainders of the original numbers or from forgetting to reduce after each multiplication. Another pitfall is trying to multiply the large numbers directly and then dividing, which is time-consuming and prone to calculation mistakes. Final Answer: The remainder when 1234 × 1235 × 1237 is divided by 8 is 6.

Question 18

A teacher multiplies 987 by a certain whole number and obtains 556781 as the product. It is later found that the digits 6 and 7 in this answer are wrong but the other digits are correct. What is the correct product?

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Introduction: This question combines number sense with error detection. The teacher multiplied 987 by an unknown whole number and wrote the product incorrectly, with exactly two wrong digits. Your task is to identify which option could be the correct product under these conditions. Given Data / Assumptions: Supposed product written by the teacher: 556781. The digits 6 and 7 in this written answer are wrong. The other digits (5, 5, 8, 1) are correct and in the correct positions. The true product must be a multiple of 987. Concept / Approach: We know: Only two digits are incorrect in the written product. Any correct answer must match 556781 in all other digit positions. The correct product must be divisible by 987. So we examine all options that differ from 556781 only in the digits 6 and 7 and check which one is divisible by 987. Step-by-Step Solution: Step 1: Compare options digit by digit. Original incorrect product: 5 5 6 7 8 1. We look for an option that keeps positions 1, 2, 5, and 6 (5, 5, 8, 1) the same and changes only the middle digits. 553681: 5 5 3 6 8 1 (two middle digits differ, but we must also check divisibility). 555181: 5 5 5 1 8 1 (two middle digits differ). 556581: 5 5 6 5 8 1 (only one of the wrong digits changed). 555681: 5 5 5 6 8 1 (two middle digits changed from 6,7 to 5,6). 557681: 5 5 7 6 8 1 (again two middle digits changed). Several options change exactly two digits, so we must use the divisibility condition next. Step 2: Check divisibility by 987. We check which candidate is an exact multiple of 987. When tested, 555681 is divisible by 987 with no remainder, while the other options do not divide exactly by 987. Thus 555681 is the only candidate that both changes exactly the wrong digits and is a valid product of 987 and some whole number. Verification / Alternative Check: Compute: 555681 / 987 = 563. So the teacher was really multiplying 987 by 563. Then recompute: 987 * 563 = 555681, which matches option 555681 exactly. Why Other Options Are Wrong: Although some options adjust two digits, they fail the divisibility test by 987 and therefore cannot be correct products of 987 and an integer. Others may implicitly change digits that were supposed to be correct. Only 555681 both respects which digits are wrong and satisfies the multiplication condition. Common Pitfalls: Students sometimes ignore the requirement that the result must be a multiple of 987 or misinterpret which digits are incorrect. Always enforce both conditions: the digit pattern and the exact divisibility by the known multiplier. Final Answer: The correct product of 987 and the whole number is 555681.

Question 19

In basic arithmetic, if 65776759 is added to 54697467, what is the rightmost digit (that is, the units place digit) of the resulting sum?

Accepted Answer

Introduction: This question checks basic number sense and your understanding of how addition works in the units place. Instead of adding the full long numbers, you only need to focus on the rightmost digits to determine the units digit of the sum quickly and efficiently. Given Data / Assumptions: The first number is 65776759. The second number is 54697467. We must find the rightmost (units place) digit of their sum. No tricks are involved beyond normal base 10 addition. Concept / Approach: In any addition problem, the units digit of the sum depends only on the units digits of the addends. Higher place values (tens, hundreds, and so on) may create carries, but the basic idea remains that the units digit is determined by adding just the units digits and then taking the units digit of that smaller sum. Step-by-Step Solution: Step 1: Identify the units digit of 65776759, which is 9. Step 2: Identify the units digit of 54697467, which is 7. Step 3: Add these two units digits: 9 + 7 = 16. Step 4: The units digit of 16 is 6, so the units digit of the full sum is also 6. Verification / Alternative check: You could add the full numbers using column addition, but you will observe that the last column still gives 9 + 7 = 16, putting 6 in the units place and carrying 1 to the tens column. This confirms that the units digit must be 6 regardless of the rest of the digits. Why Other Options Are Wrong: 4: This would require 9 + 7 to end in 4, which is not true. 9: No combination of 9 and 7 in simple addition yields a units digit of 9. 0: 9 + 7 does not produce a units digit of 0. 8: This also does not match the correct units digit from 9 + 7. Common Pitfalls: Some learners mistakenly try to add the entire numbers, which is time consuming and increases the chance of arithmetic errors. Others confuse the carry and the units digit itself. Remember that only the units digits 9 and 7 matter for this type of question. Final Answer: Therefore, the rightmost digit of 65776759 + 54697467 is 6.

Question 20

A box contains 90 screws each weighing 100 g and 100 bolts each weighing 150 g. If the total weight of the box together with all the screws and bolts is 35.5 kg, what is the weight of the empty box alone?

Accepted Answer

Introduction: This problem tests your understanding of unit conversion between grams and kilograms and your ability to separate the weight of contents from the weight of a container. It is a typical aptitude question on arithmetic and averages in the context of weights and measures. Given Data / Assumptions: Number of screws = 90, each of weight 100 g. Number of bolts = 100, each of weight 150 g. Total weight of box plus all contents = 35.5 kg. We assume 1000 g = 1 kg. The weight of the box is constant and does not vary with the contents. Concept / Approach: We first compute the total weight of screws and bolts in kilograms. Then we subtract that total contents weight from the combined weight (box plus contents) to find the weight of the empty box. This is a straightforward application of addition, multiplication, and subtraction with unit conversion. Step-by-Step Solution: Step 1: Total weight of screws = 90 * 100 g = 9000 g. Step 2: Total weight of bolts = 100 * 150 g = 15000 g. Step 3: Total weight of all items in grams = 9000 g + 15000 g = 24000 g. Step 4: Convert 24000 g to kg: 24000 / 1000 = 24 kg. Step 5: Total given weight (box + items) = 35.5 kg. Step 6: Weight of empty box = 35.5 kg - 24 kg = 11.5 kg. Verification / Alternative check: If the box alone weighs 11.5 kg, then box plus items should weigh 11.5 kg + 24 kg = 35.5 kg, which matches the given total. This confirms that our calculation is consistent and correct. Why Other Options Are Wrong: 10.5 kgs: Gives total weight 10.5 + 24 = 34.5 kg, not 35.5 kg. 12 kgs: Gives total weight 12 + 24 = 36 kg, which is too high. 9.6 kgs: Gives total weight 9.6 + 24 = 33.6 kg, which is too low. 8.5 kgs: Also does not match the required total when added to 24 kg. Common Pitfalls: A common mistake is forgetting to convert grams into kilograms or mixing up 150 g as 0.15 kg instead of 0.150 kg. Others may incorrectly add the numbers or subtract in the wrong direction. Always keep units consistent and carefully distinguish between box weight and contents weight. Final Answer: The weight of the empty box is 11.5 kgs.

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