Q: What is the units digit of 7^105?

Concept / Approach Units digits of powers of 7 cycle every 4: 7, 9, 3, 1, then repeat. Step-by-step calculation Compute exponent modulo 4: 105 mod 4 = 1Position 1 in the cycle corresponds to units digit 7. Common pitfall Using 105 ÷ 4 = 26.25 and rounding incorrectly. Only the remainder matters. Final Answer 7

Q: (1000)9 ? 1024 = ?

Cannot be determined from the given information

Q: Compute: 106^2 − 94^2

Concept / Approach Use the difference of squares: a^2 − b^2 = (a − b)(a + b). Step-by-step calculation a = 106, b = 94a − b = 12a + b = 200(a − b)(a + b) = 12 × 200 = 2400 Quick check Both 106^2 and 94^2 end with 36; their difference should be a multiple of 100, consistent with 2400. Final Answer 2400

Q: Find the value of 112 × 54

Concept / Approach Use distributive multiplication to keep arithmetic clean. Step-by-step calculation 112 × 54 = 112 × (50 + 4)= 112 × 50 + 112 × 4= 5600 + 448= 6048 Quick check Approximate 112 × 54 ≈ 110 × 54 = 5940; exact 6048 is close and reasonable. Final Answer 6048

Q: Find the value of 587 × 999

Concept / Approach Use 999 = 1000 − 1 to convert into an easy subtraction. Step-by-step calculation 587 × 999 = 587 × (1000 − 1)= 587000 − 587= 586413 Quick check Multiplying by 999 is just 1000× minus the number itself; 587000 − 587 = 586413 is consistent. Final Answer 586413

Q: Evaluate the sum: 51 + 52 + 53 + … + 100

Concept / Approach This is an arithmetic series with first term 51, last term 100, and common difference 1. Use the sum formula Sn = n(First + Last) / 2 where n = 100 − 51 + 1 = 50. Step-by-step calculation n = 50S = 50 × (51 + 100) / 2= 50 × 151 / 2= 25 × 151 = 3775 Verification Pairing terms: (51 + 100), (52 + 99), … each pair sums to 151; there are 25 pairs ⇒ 25 × 151 = 3775. Common pitfalls Using 50 × 151 (forgetting to divide by 2) or taking the wrong number of terms. Final Answer 3775

Question 1

What is the largest four-digit number that is exactly divisible by 88?

Accepted Answer

Concept / Approach The largest four-digit number is 9999. To find the largest number divisible by 88 not exceeding 9999, compute floor(9999 ÷ 88) and multiply back by 88. Step-by-step calculation 9999 ÷ 88 = 113.625…Largest integer ≤ 113.625 is 113Required multiple = 113 × 88 = 9944 Quick check 88 × 114 = 10032 > 9999, so 9944 is indeed the greatest four-digit multiple. Common pitfall Do not round 113.625 up to 114; we must take the floor to stay ≤ 9999. Final Answer 9944

Question 2

Which of the following numbers is a prime number?

Accepted Answer

173

Question 3

In a division problem with zero remainder, a student used 12 as the divisor instead of 21 and got a quotient of 35. What is the correct quotient?

Accepted Answer

Given data True division has remainder 0. Correct divisor = 21. Student mistakenly used divisor = 12 and obtained quotient = 35. Concept / Approach If the true remainder is 0, the dividend must be an exact multiple of the correct divisor (21). The student's calculation tells us the dividend also equals 12 × 35 (since a quotient of 35 was obtained). Step-by-step calculation Dividend from mistaken work = 12 × 35 = 420Check with the true divisor: 420 ÷ 21 = 20 (remainder 0) Verification Since the problem states the remainder is 0 in the correct setup, 420 being exactly divisible by 21 confirms consistency. Common pitfalls Forgetting that a zero remainder forces the dividend to be a multiple of the correct divisor. Trying to adjust the quotient proportionally without first finding the actual dividend. Final Answer 20

Question 4

A number leaves a remainder of 39 when divided by 357. What is the remainder when the same number is divided by 17?

Accepted Answer

Given data N ≡ 39 (mod 357) Find N mod 17 Concept / Approach Use modular arithmetic. If 357 is a multiple of 17, then N ≡ 39 (mod 17). Step-by-step calculation 357 = 17 × 21 ⇒ 357 ≡ 0 (mod 17)Therefore, N ≡ 39 (mod 17)39 ÷ 17 = 2 remainder 5 ⇒ 39 ≡ 5 (mod 17) Quick check Add or subtract multiples of 17 from 39: 39 − 34 = 5. Final Answer 5

Question 5

In a division problem, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 46, what is the dividend?

Accepted Answer

Given data Divisor d = 10 × quotient q Divisor d = 5 × remainder r Remainder r = 46 Concept / Approach From d = 5r, find d. Then from d = 10q, find q. Finally use Dividend = dq + r. Step-by-step calculation d = 5r = 5 × 46 = 230d = 10q ⇒ q = d / 10 = 230 / 10 = 23Dividend = dq + r = 230 × 23 + 46 = 5290 + 46 = 5336 Validity check Remainder r = 46 < d = 230, which is valid. Final Answer 5336

Question 6

What is the units digit of 7^105?

Accepted Answer

Concept / Approach Units digits of powers of 7 cycle every 4: 7, 9, 3, 1, then repeat. Step-by-step calculation Compute exponent modulo 4: 105 mod 4 = 1Position 1 in the cycle corresponds to units digit 7. Common pitfall Using 105 ÷ 4 = 26.25 and rounding incorrectly. Only the remainder matters. Final Answer 7

Question 7

(1000)9 ? 1024 = ?

Accepted Answer

Cannot be determined from the given information

Question 8

Compute: 106^2 − 94^2

Accepted Answer

Concept / Approach Use the difference of squares: a^2 − b^2 = (a − b)(a + b). Step-by-step calculation a = 106, b = 94a − b = 12a + b = 200(a − b)(a + b) = 12 × 200 = 2400 Quick check Both 106^2 and 94^2 end with 36; their difference should be a multiple of 100, consistent with 2400. Final Answer 2400

Question 9

Find the value of 112 × 54

Accepted Answer

Concept / Approach Use distributive multiplication to keep arithmetic clean. Step-by-step calculation 112 × 54 = 112 × (50 + 4)= 112 × 50 + 112 × 4= 5600 + 448= 6048 Quick check Approximate 112 × 54 ≈ 110 × 54 = 5940; exact 6048 is close and reasonable. Final Answer 6048

Question 10

Find the value of 587 × 999

Accepted Answer

Concept / Approach Use 999 = 1000 − 1 to convert into an easy subtraction. Step-by-step calculation 587 × 999 = 587 × (1000 − 1)= 587000 − 587= 586413 Quick check Multiplying by 999 is just 1000× minus the number itself; 587000 − 587 = 586413 is consistent. Final Answer 586413

Question 11

Evaluate the sum: 51 + 52 + 53 + … + 100

Accepted Answer

Concept / Approach This is an arithmetic series with first term 51, last term 100, and common difference 1. Use the sum formula Sn = n(First + Last) / 2 where n = 100 − 51 + 1 = 50. Step-by-step calculation n = 50S = 50 × (51 + 100) / 2= 50 × 151 / 2= 25 × 151 = 3775 Verification Pairing terms: (51 + 100), (52 + 99), … each pair sums to 151; there are 25 pairs ⇒ 25 × 151 = 3775. Common pitfalls Using 50 × 151 (forgetting to divide by 2) or taking the wrong number of terms. Final Answer 3775

Question 12

What is the units digit of (795 − 358)?

Accepted Answer

Concept / Approach The units digit of a difference depends only on the units digits (with borrowing if needed). Step-by-step calculation Compute 795 − 358:Units: 5 − 8 requires borrowing ⇒ 15 − 8 = 7Tens: (8 − 5) = 3 after borrowingHundreds: 7 − 3 = 4Result = 437 Final Answer 7

Question 13

What is the smallest five-digit number that is exactly divisible by 41?

Accepted Answer

Concept / Approach Let the smallest five-digit number be 10000. Find the least multiple of 41 that is ≥ 10000. Step-by-step calculation 10000 ÷ 41 = 243.902… ⇒ take ceiling to 244Required multiple = 244 × 41 = 10004 Verification 243 × 41 = 9963 < 10000; 244 × 41 = 10004 ≥ 10000. Final Answer 10004

Question 14

Solve for x: 4500 × x = 3375

Accepted Answer

Concept / Approach Isolate x by dividing both sides by 4500. Step-by-step calculation x = 3375 ÷ 4500= (45 × 75) ÷ (60 × 75) = 45 ÷ 60 = 3 ÷ 4 = 0.75 Verification 4500 × 0.75 = 3375 ✔ Common pitfalls Not fully reducing the fraction 3375/4500. Final Answer 0.75

Question 15

Find the value that satisfies: x + 3699 + 1985 − 2047 = 31111

Accepted Answer

Concept / Approach Group constants and move them to the right-hand side. Step-by-step calculation x + (3699 + 1985 − 2047) = 31111x + 3637 = 31111x = 31111 − 3637 = 27474 Verification 27474 + 3699 + 1985 − 2047 = 27474 + 3637 = 31111 ✔ Final Answer 27474

Question 16

Compute: 72519 × 9999

Accepted Answer

Concept / Approach Use 9999 = 10000 − 1 to convert the product into a simple subtraction. Step-by-step calculation 72519 × 9999 = 72519 × (10000 − 1)= 72519 × 10000 − 72519= 725190000 − 72519= 725117481 Quick check Multiplying by 9999 is like taking 10000× and subtracting the number once. Final Answer 725117481

Question 17

The difference of the squares of two consecutive odd integers is divisible by which integer?

Accepted Answer

Concept / Approach Let the consecutive odd integers be (2k + 1) and (2k + 3). Use the identity a^2 − b^2 = (a − b)(a + b). Step-by-step derivation (2k + 3)^2 − (2k + 1)^2 = [(2k + 3) − (2k + 1)] × [(2k + 3) + (2k + 1)]= (2) × (4k + 4) = 8(k + 1) Conclusion The expression is always a multiple of 8 (but not always 16), hence divisible by 8. Final Answer 8

Question 18

A number gives remainders 4, 8, and 12 when successively divided by 5, 9, and 13 (short division for 585 = 5 × 9 × 13). What remainder would occur if divided directly by 585?

Accepted Answer

Given data 585 = 5 × 9 × 13 with pairwise coprime factors. Short-division remainders: 4 (mod 5), 8 (mod 9), 12 (mod 13). Concept / Approach Chinese Remainder Theorem (CRT): find r such that r ≡ 4 (mod 5), r ≡ 8 (mod 9), r ≡ 12 (mod 13) Step-by-step solution Start with r ≡ 8 (mod 9) ⇒ r = 8 + 9aImpose r ≡ 12 (mod 13): 8 + 9a ≡ 12 (mod 13)9a ≡ 4 (mod 13)Inverse of 9 mod 13 is 3 (since 9 × 3 = 27 ≡ 1)a ≡ 3 × 4 = 12 (mod 13) ⇒ a = 12 + 13tThus r = 8 + 9(12 + 13t) = 116 + 117tNow impose r ≡ 4 (mod 5): 116 + 117t ≡ 4 (mod 5)116 ≡ 1, 117 ≡ 2 (mod 5) ⇒ 1 + 2t ≡ 42t ≡ 3 (mod 5) ⇒ multiply by inverse of 2 (which is 3): t ≡ 9 ≡ 4 (mod 5)Take t = 4 ⇒ r = 116 + 117 × 4 = 584 Verification 584 mod 5 = 4, 584 mod 9 = 8, 584 mod 13 = 12. Common pitfalls Confusing successive short-division remainders with non-coprime moduli or forgetting CRT conditions. Final Answer 584

Question 19

Find the sum of all two-digit numbers that are divisible by 5

Accepted Answer

Concept / Approach Two-digit multiples of 5 are 10, 15, …, 95 (arithmetic sequence with d = 5). Use the sum formula for an arithmetic series. Step-by-step calculation First = 10, Last = 95, d = 5Number of terms n = ((95 − 10)/5) + 1 = 18Sum S = n(First + Last)/2 = 18 × (10 + 95)/2= 18 × 105 / 2 = 9 × 105 = 945 Verification Pair terms (10 + 95), (15 + 90), … each = 105; 9 pairs ⇒ 9 × 105 = 945. Final Answer 945

Question 20

Evaluate: (35423 + 7164 + 41720) − (317 × 89)

Accepted Answer

Concept / Approach Add the three numbers, compute the product, then subtract. Step-by-step calculation Sum = 35423 + 7164 + 41720 = 84307Product = 317 × 89 = 317 × (90 − 1) = 28530 − 317 = 28213Result = 84307 − 28213 = 56094 Quick check Estimate: 84k − 28k ≈ 56k, matching 56094. Final Answer 56094

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