Q: Which of the following numbers divides (3^25 + 3^26 + 3^27 + 3^28) exactly?

Given data Expression: 3^25 + 3^26 + 3^27 + 3^28 Concept / Approach Factor out the smallest power to reveal a simple integer factor. Step-by-step calculation 3^25 + 3^26 + 3^27 + 3^28 = 3^25(1 + 3 + 3^2 + 3^3)= 3^25(1 + 3 + 9 + 27) = 3^25 × 40Therefore, any divisor of 40 divides the entire sum; in particular, 40 divides it. Why others do not 7, 11, 13 are not factors of 40 and do not divide any power of 3, hence they do not necessarily divide the expression. Final Answer Divisible by 40.

Q: Evaluate (489 + 375)^2 − (489 − 375)^2.

Given data Compute (x + y)^2 − (x − y)^2 for x = 489, y = 375. Concept / Approach Identity: (x + y)^2 − (x − y)^2 = 4xy (difference of squares of conjugates). Step-by-step calculation x·y = 489 × 375 = (500 − 11) × 375 = 187,500 − 4,125 = 183,3754xy = 4 × 183,375 = 733,500 Verification (489 + 375) = 864, (489 − 375) = 114; 864^2 − 114^2 = (864 − 114)(864 + 114) = 750 × 978 = 733,500. Final Answer 733500

Q: A number leaves remainder 29 when divided by 56. What is the remainder when it is divided by 8?

Given data N ≡ 29 (mod 56) Concept / Approach If m is a multiple of n, then N ≡ r (mod m) ⇒ N ≡ r (mod n). Here, 56 is a multiple of 8. Step-by-step calculation N = 56k + 29 ⇒ N mod 8 = (56k mod 8) + (29 mod 8)56k mod 8 = 0; 29 mod 8 = 5Therefore remainder = 5. Common pitfalls Forgetting to reduce 29 modulo 8. Final Answer 5

Q: Find a common factor of (4^743 + 4^343) and (4^747 + 4^347).

Given data S1 = 4^743 + 4^343 S2 = 4^747 + 4^347 Concept / Approach Factor out common powers of 4 and compare. Step-by-step factorization S1 = 4^343(4^400 + 1)S2 = 4^347(4^400 + 1) = 4^343 · 4^4 · (4^400 + 1)Thus, 4^343(4^400 + 1) divides both S1 and S2. Why alternatives fail 4^343(4^400 − 1): the (4^400 − 1) factor is not present in either expression. 4^344(4^400 + 1): divides S2 but not S1 (S1 has only 4^343). 4^400 − 1: unrelated to the given sums. Final Answer Common factor = 4^343(4^400 + 1).

Q: If 517*324 is divisible by 3, find the smallest digit that can replace .

Given data Number: 5,1,7,,3,2,4 (seven digits with unknown *) Divisibility target: by 3 Concept / Approach A number is divisible by 3 iff the sum of its digits is a multiple of 3. Step-by-step calculation Sum (excluding *): 5 + 1 + 7 + 3 + 2 + 4 = 22We need 22 + * ≡ 0 (mod 3) ⇒ * ≡ 2 (mod 3)Smallest digit satisfying this is 2. Verification 22 + 2 = 24, which is divisible by 3. Final Answer 2

Q: Which of the following is a prime number?

Given data & approach Test each option for primality by trial division up to its square root. Checks 221 = 13 × 17 ⇒ composite217 = 7 × 31 ⇒ composite223: not divisible by 2, 3 (2+2+3=7), 5, 7 (7×32=224), 11 (11×20=220), 13 (13×17=221) ⇒ prime253 = 11 × 23 ⇒ composite Final Answer 223 is prime.

Q: The difference between a positive proper fraction and its reciprocal is 9/20. Find the fraction.

Given data Let the proper fraction be x, 0 1). Difference: (1/x − x) = 9/20. Concept / Approach Solve the quadratic arising from the equation. Step-by-step calculation 1/x − x = 9/20Multiply by 20x: 20 − 20x^2 = 9xRearrange: 20x^2 + 9x − 20 = 0Discriminant Δ = 9^2 + 4·20·20 = 81 + 1600 = 1681 = 41^2x = (−9 ± 41) / 40 ⇒ positive root x = 32/40 = 4/5 Verification 1/x − x = 5/4 − 4/5 = (25 − 16)/20 = 9/20 ✔ Common pitfalls Using x − 1/x instead of 1/x − x (the latter is positive for a proper fraction). Final Answer The fraction is 4/5.

Question 1

Find the remainder when 17,200 is divided by 18.

Accepted Answer

Given data Dividend: 17,200 Divisor: 18 Concept / Approach Use either direct division or modular reasoning with coprime factors: 18 = 2 × 9. We need an r such that 0 ≤ r < 18, r ≡ 0 (mod 2) and r ≡ 17,200 (mod 9). Step-by-step calculation (direct) 18 × 955 = 17,190Remainder r = 17,200 − 17,190 = 10 Step-by-step (mod reasoning) 17,200 mod 9: digit sum = 1+7+2 = 10 ⇒ 10 mod 9 = 1 ⇒ r ≡ 1 (mod 9)Also r must be even (since 17,200 is even).Even residues < 18 that are ≡ 1 (mod 9): {10}. Hence r = 10. Common pitfalls Confusing remainder with quotient. Forgetting the remainder must be less than the divisor. Final Answer Remainder = 10.

Question 2

Which of the following numbers divides (3^25 + 3^26 + 3^27 + 3^28) exactly?

Accepted Answer

Given data Expression: 3^25 + 3^26 + 3^27 + 3^28 Concept / Approach Factor out the smallest power to reveal a simple integer factor. Step-by-step calculation 3^25 + 3^26 + 3^27 + 3^28 = 3^25(1 + 3 + 3^2 + 3^3)= 3^25(1 + 3 + 9 + 27) = 3^25 × 40Therefore, any divisor of 40 divides the entire sum; in particular, 40 divides it. Why others do not 7, 11, 13 are not factors of 40 and do not divide any power of 3, hence they do not necessarily divide the expression. Final Answer Divisible by 40.

Question 3

Evaluate (489 + 375)^2 − (489 − 375)^2.

Accepted Answer

Given data Compute (x + y)^2 − (x − y)^2 for x = 489, y = 375. Concept / Approach Identity: (x + y)^2 − (x − y)^2 = 4xy (difference of squares of conjugates). Step-by-step calculation x·y = 489 × 375 = (500 − 11) × 375 = 187,500 − 4,125 = 183,3754xy = 4 × 183,375 = 733,500 Verification (489 + 375) = 864, (489 − 375) = 114; 864^2 − 114^2 = (864 − 114)(864 + 114) = 750 × 978 = 733,500. Final Answer 733500

Question 4

A number leaves remainder 29 when divided by 56. What is the remainder when it is divided by 8?

Accepted Answer

Given data N ≡ 29 (mod 56) Concept / Approach If m is a multiple of n, then N ≡ r (mod m) ⇒ N ≡ r (mod n). Here, 56 is a multiple of 8. Step-by-step calculation N = 56k + 29 ⇒ N mod 8 = (56k mod 8) + (29 mod 8)56k mod 8 = 0; 29 mod 8 = 5Therefore remainder = 5. Common pitfalls Forgetting to reduce 29 modulo 8. Final Answer 5

Question 5

Find a common factor of (4^743 + 4^343) and (4^747 + 4^347).

Accepted Answer

Given data S1 = 4^743 + 4^343 S2 = 4^747 + 4^347 Concept / Approach Factor out common powers of 4 and compare. Step-by-step factorization S1 = 4^343(4^400 + 1)S2 = 4^347(4^400 + 1) = 4^343 · 4^4 · (4^400 + 1)Thus, 4^343(4^400 + 1) divides both S1 and S2. Why alternatives fail 4^343(4^400 − 1): the (4^400 − 1) factor is not present in either expression. 4^344(4^400 + 1): divides S2 but not S1 (S1 has only 4^343). 4^400 − 1: unrelated to the given sums. Final Answer Common factor = 4^343(4^400 + 1).

Question 6

If 517*324 is divisible by 3, find the smallest digit that can replace .

Accepted Answer

Given data Number: 5,1,7,,3,2,4 (seven digits with unknown *) Divisibility target: by 3 Concept / Approach A number is divisible by 3 iff the sum of its digits is a multiple of 3. Step-by-step calculation Sum (excluding *): 5 + 1 + 7 + 3 + 2 + 4 = 22We need 22 + * ≡ 0 (mod 3) ⇒ * ≡ 2 (mod 3)Smallest digit satisfying this is 2. Verification 22 + 2 = 24, which is divisible by 3. Final Answer 2

Question 7

Which of the following is a prime number?

Accepted Answer

Given data & approach Test each option for primality by trial division up to its square root. Checks 221 = 13 × 17 ⇒ composite217 = 7 × 31 ⇒ composite223: not divisible by 2, 3 (2+2+3=7), 5, 7 (7×32=224), 11 (11×20=220), 13 (13×17=221) ⇒ prime253 = 11 × 23 ⇒ composite Final Answer 223 is prime.

Question 8

The difference between a positive proper fraction and its reciprocal is 9/20. Find the fraction.

Accepted Answer

Given data Let the proper fraction be x, 0 < x < 1. Then its reciprocal is 1/x (which > 1). Difference: (1/x − x) = 9/20. Concept / Approach Solve the quadratic arising from the equation. Step-by-step calculation 1/x − x = 9/20Multiply by 20x: 20 − 20x^2 = 9xRearrange: 20x^2 + 9x − 20 = 0Discriminant Δ = 9^2 + 4·20·20 = 81 + 1600 = 1681 = 41^2x = (−9 ± 41) / 40 ⇒ positive root x = 32/40 = 4/5 Verification 1/x − x = 5/4 − 4/5 = (25 − 16)/20 = 9/20 ✔ Common pitfalls Using x − 1/x instead of 1/x − x (the latter is positive for a proper fraction). Final Answer The fraction is 4/5.

Question 9

Compute: −84 × 29 + 365

Accepted Answer

Given data & approach Evaluate the linear combination carefully observing the sign. Step-by-step arithmetic 84 × 29 = 84 × (30 − 1) = 2520 − 84 = 2436−84 × 29 = −2436−2436 + 365 = −2071 Common pitfalls Sign errors when distributing the negative multiplier. Final Answer −2071

Question 10

Given that (2^32 + 1) is divisible by some whole number N, which of the following is also divisible by N?

Accepted Answer

Given data Let a = 2^32. We are told N | (a + 1). Concept / Approach If N divides (a + 1), then for any odd k: N | (a^k + 1) because a ≡ −1 (mod N) ⇒ a^k ≡ (−1)^k. Step-by-step reasoning a ≡ −1 (mod N)For k = 3 (odd): a^3 + 1 ≡ (−1)^3 + 1 = −1 + 1 = 0 (mod N)Thus N divides a^3 + 1 = (2^32)^3 + 1 = 2^96 + 1. Why others need not work 2^64 + 1 = a^2 + 1 ≡ 1 + 1 = 2 (mod N), not generally 0 since N is odd (2^32 + 1 is odd). 2^16 + 1 and 2^48 + 1 are not of the form a^k ± 1 with k odd in terms of a = 2^32, hence no guaranteed divisibility by N. Final Answer The expression 2^96 + 1 is also divisible by N.

Question 11

Find ?: 3251 + 587 + 369 − ? = 3007

Accepted Answer

Given data Equation: 3251 + 587 + 369 − ? = 3007 Concept / Approach First sum the known addends; then solve for the unknown by transposition. Step-by-step calculation S = 3251 + 587 = 3838S = 3838 + 369 = 42074207 − ? = 3007 ⇒ ? = 4207 − 3007? = 1200 Verification / Alternative Plug back: 3251 + 587 + 369 − 1200 = 4207 − 1200 = 3007 ✔ Common pitfalls Subtracting in the wrong order (3007 − 4207) or mis-adding 587 and 369. Final Answer 1200

Question 12

Find the sum of the first 45 natural numbers.

Accepted Answer

Given data Needed: S = 1 + 2 + … + 45 Concept / Approach Use the standard formula for the first n natural numbers: S = n(n + 1)/2. Step-by-step calculation S = 45 × 46 / 2= 45 × 23= (40 × 23) + (5 × 23) = 920 + 115 = 1035 Verification / Alternative Pairing: (1+45), (2+44), … ⇒ 22 pairs of 46 plus the middle 23 ⇒ 22×46 + 23 = 1012 + 23 = 1035 ✔ Common pitfalls Using n(n − 1)/2 by mistake (that sums 1 to n−1). Final Answer 1035

Question 13

Find the difference between the place values of the two 7s in 69758472.

Accepted Answer

Given data Number: 69758472 (8 digits) Two 7s: positions from left are 3rd and 7th digits. Concept / Approach Place value = digit × (place weight). From right, weights are 1, 10, 100, 1000, … Step-by-step identification Digits (with place): 6(10^7), 9(10^6), 7(10^5), 5(10^4), 8(10^3), 4(10^2), 7(10^1), 2(10^0)Place value of left 7: 7 × 10^5 = 700000Place value of right 7: 7 × 10^1 = 70Difference = 700000 − 70 = 699930 Verification Both place values computed from positional weights; subtraction straightforward. Common pitfalls Confusing face value (7) with place value (e.g., 700000). Off-by-one in counting positions from the right. Final Answer 699930

Question 14

If 653xy is divisible by 90, find x + y.

Accepted Answer

Given data Five-digit number: 6,5,3,x,y; divisible by 90. Concept / Approach 90 = 9 × 10; a number divisible by 90 must be divisible by both 9 and 10. Divisible by 10 ⇒ last digit y = 0. Divisible by 9 ⇒ sum of digits multiple of 9. Step-by-step calculation Digit sum = 6 + 5 + 3 + x + y = 14 + x + yWith y = 0 ⇒ 14 + x ≡ 0 (mod 9) ⇒ x ≡ −14 ≡ 4 (mod 9)Valid digit x = 4; hence x + y = 4 + 0 = 4 Verification 65340 ÷ 90 = 726 exactly; digit sum 6+5+3+4+0=18 (multiple of 9) and ends with 0. Common pitfalls Forgetting the divisibility-by-10 condition (y must be 0). Final Answer 4

Question 15

What is the smallest three-digit prime number?

Accepted Answer

Given data & approach Three-digit numbers start at 100. Check primality from 100 upward. Step-by-step reasoning 100 is even ⇒ composite.101 has no divisor among primes ≤ √101 ≈ 10 (i.e., 2, 3, 5, 7): not divisible by any ⇒ prime.Therefore the smallest three-digit prime is 101. Verification Next candidates 102 (even), 103 (prime but larger) confirm minimality. Common pitfalls Choosing 103 (also prime) but not the smallest. Assuming 111 is prime (actually 3 × 37). Final Answer 101

Question 16

Find the difference between the place value and the face value of 6 in 856973.

Accepted Answer

Given data Number: 856973; digit of interest: 6 Concept / Approach Face value of a digit is the digit itself. Place value = digit × (place weight). Here, 856973 = 8(10^5) + 5(10^4) + 6(10^3) + 9(10^2) + 7(10^1) + 3. Step-by-step calculation Place value of 6: 6 × 1000 = 6000Face value of 6: 6Difference = 6000 − 6 = 5994 Verification The 6 is in the thousands place (third from the right): weight 10^3. Common pitfalls Using hundreds (10^2) by miscounting positions. Final Answer 5994

Question 17

Which of the following cannot be the square of a natural number?

Accepted Answer

Given data & approach Check each option against properties of perfect squares. Checks 1936 = 44^2 ✔2209 = 47^2 ✔2401 = 49^2 ✔1537 ends with 7. No perfect square in base-10 ends with 2, 3, 7 or 8 ⇒ cannot be a perfect square. Additional note Also, √1537 ≈ 39.2; 39^2 = 1521 and 40^2 = 1600, neither equals 1537. Final Answer Number that cannot be a square: 1537.

Question 18

Find the unit digit of (365 × 659 × 771).

Accepted Answer

Given data & approach Unit digit of a product depends only on unit digits of factors. Step-by-step calculation Units(365) = 5; Units(659) = 9; Units(771) = 1Compute: (5 × 9) = 45 ⇒ units 5Then 5 × 1 ⇒ units 5 Verification Any number ending in 5 times any odd number ends in 5; multiplying by 1 keeps the unit digit unchanged. Common pitfalls Multiplying full numbers unnecessarily; only unit digits matter. Final Answer 5

Question 19

How many three-digit numbers are divisible by 6?

Accepted Answer

Given data Three-digit range: 100 to 999 (inclusive). Divisible by 6 ⇔ divisible by 2 and 3. Concept / Approach Count multiples of 6 between 100 and 999 using arithmetic progression count or floor-division method. Step-by-step calculation Smallest 3-digit multiple: 102 = 6×17Largest 3-digit multiple: 996 = 6×166Count = ((996 − 102)/6) + 1 = (894/6) + 1 = 149 + 1 = 150 Alternative check ⌊999/6⌋ − ⌊99/6⌋ = 166 − 16 = 150 ✔ Common pitfalls Forgetting the +1 in the term-count formula or mistakenly including 100/999. Final Answer 150

Question 20

Find the smallest six-digit number exactly divisible by 111.

Accepted Answer

Given data & approach We want the least multiple of 111 that is ≥ 100000 (smallest six-digit number). Step-by-step calculation Compute the ceiling multiple: ⌈100000 / 111⌉111 × 900 = 99900111 × 901 = 100011 (≥ 100000)Thus the smallest six-digit multiple is 100011. Verification 100011 ÷ 111 = 901 exactly; 100110 ÷ 111 = 902 with remainder −12; 100101 is not an exact multiple; 111000 is a multiple (×1000) but larger than 100011. Common pitfalls Taking 111000 as the smallest just because it is a tidy multiple; it is not the least six-digit multiple. Final Answer 100011

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