Q: Evaluate the square: 1397 × 1397.

Problem Restatement Calculate 13972 quickly without full long multiplication. Concept / Approach Use the identity (a − b)2 = a2 − 2ab + b2 with a = 1400 and b = 3. Step-by-Step Calculation 1397 = 1400 − 3(1400 − 3)2 = 14002 − 2 × 1400 × 3 + 32= 1,960,000 − 8,400 + 9 = 1,951,609 Verification Sanity check: 14002 = 1,960,000; the square should be slightly less — difference 8,391, consistent. Final Answer 1951609

Q: Evaluate: {(476 + 424) 2 − 4 × 476 × 424}.

Problem Restatement Compute {(476 + 424)2 − 4 × 476 × 424} using algebraic identities. Concept / Approach Recognize the identity: (a + b)2 − 4ab = a2 + 2ab + b2 − 4ab = a2 − 2ab + b2 = (a − b)2. Step-by-Step Calculation Let a = 476, b = 424(a − b)2 = (476 − 424)2 = 522 = 2704 Common Pitfalls Attempting brute-force squaring; spotting the identity is faster and reduces errors. Final Answer 2704

Q: If 64 2 − 36 2 = 20 × x, find the value of x.

Problem Restatement Use the difference-of-squares identity to evaluate 642 − 362 and solve for x in 20x. Concept / Approach (A2 − B2) = (A − B)(A + B). Step-by-Step Calculation 642 − 362 = (64 − 36)(64 + 36) = 28 × 100 = 28002800 = 20x → x = 2800 ÷ 20 = 140 Final Answer 140

Q: If n is a natural number, then the expression (6n 2 + 6n) is always divisible by which of the following?

Problem Restatement Find the highest guaranteed divisor among the choices for 6n2 + 6n when n is any natural number. Given / Factorization 6n2 + 6n = 6n(n + 1) Concept / Approach Among two consecutive integers n and (n + 1), one is even. Therefore n(n + 1) is even → divisible by 2. Since there is an explicit factor 6 = 2 × 3, the whole product has at least 2 × 2 × 3 = 12. Step-by-Step Reasoning 6n(n + 1) = (2 × 3) × n(n + 1)n(n + 1) contributes at least one factor 2Hence overall divisibility includes 2 × 2 × 3 = 12 Common Pitfalls Marking 6 as the answer; while true, 12 is also always a divisor and is larger. Assuming divisibility by 18; not guaranteed because an extra factor of 3 is not assured for all n. Final Answer 12

Q: Which of the following numbers will exactly divide (49 15 − 1)?

Problem Restatement Test the divisibility of 4915 − 1 by the given options using number-theory facts. Concept / Approach For any integer a, an − 1 is always divisible by (a − 1). If n is even, an − 1 is also divisible by (a + 1); here n = 15 (odd), so this does not apply. Step-by-Step Reasoning a = 49, n = 15 → a − 1 = 48Thus 4915 − 1 is divisible by 48.Check a + 1 = 50: 49 ≡ −1 (mod 50) → 4915 − 1 ≡ (−1)15 − 1 = −2 ≠ 0 (mod 50) Final Answer 48

Q: Find the unit digit of the product: 784 × 618 × 917 × 463.

Problem Restatement Determine only the units digit (last digit) of the large product 784 × 618 × 917 × 463. Concept / Approach The unit digit of a product depends only on the unit digits of the factors. Step-by-Step Calculation Consider unit digits: 784 (→ 4), 618 (→ 8), 917 (→ 7), 463 (→ 3)Multiply sequentially (keep only unit digit at each step):4 × 8 = 32 → unit digit 22 × 7 = 14 → unit digit 44 × 3 = 12 → unit digit 2 Common Pitfalls Unnecessary full multiplication; tracking only unit digits suffices. Final Answer 2

Q: Which natural number nearest to 9217 is exactly divisible by 88?

Problem Restatement Find the closest multiple of 88 to 9217. Concept / Approach Compute the nearest multiples by division and compare distances. Step-by-Step Calculation Compute 9217 ÷ 88.88 × 104 = 9152 (since 88 × 100 = 8800; 8800 + 352 = 9152)Next multiple: 88 × 105 = 9240Distances: |9217 − 9152| = 65; |9240 − 9217| = 23Nearest multiple is 9240 Quick Check 9216 is not a multiple of 88 (88 × 104 = 9152; +64 would be 9216, but 64 is not 88). Final Answer 9240

Question 1

Evaluate: 8988 ÷ 8 ÷ 4 (perform divisions from left to right). What is the value?

Accepted Answer

Problem Restatement Compute the numerical value of the expression 8988 ÷ 8 ÷ 4 by following the left-to-right order for division. Given data Expression: 8988 ÷ 8 ÷ 4 Order of operations: division is performed left → right. Concept / Approach In an expression containing only division (and/or multiplication), evaluate strictly from left to right. Do not regroup unless parentheses are given. Step-by-Step Calculation First division: 8988 ÷ 8 = 1123.5Second division: 1123.5 ÷ 4 = 280.875 Verification Direct reduction: 8988 ÷ (8 × 4) = 8988 ÷ 32 = 280.875 — same result. Common Pitfalls Evaluating as 8988 ÷ (8 ÷ 4) = 8988 ÷ 2 = 4494 (incorrect because no parentheses change the natural left-to-right order). Final Answer 280.875

Question 2

Compute the product: 5358 × 51.

Accepted Answer

Problem Restatement Multiply 5358 by 51 efficiently. Given data Operands: 5358 and 51 Concept / Approach Use distributive property: 51 = 50 + 1. Step-by-Step Calculation 5358 × 51 = 5358 × (50 + 1)= 5358 × 50 + 5358 × 1= 267900 + 5358 = 273258 Verification Standard multiplication gives the same sum of partial products (5358 × 50 and 5358 × 1). Final Answer 273258

Question 3

The difference between two numbers is 1365. Dividing the larger by the smaller gives quotient 6 and remainder 15. What is the smaller number?

Accepted Answer

Problem Restatement Two numbers (larger L, smaller S) satisfy: L − S = 1365 and L = 6S + 15. Find S. Given data Difference: L − S = 1365 Division algorithm: L = 6S + 15 (quotient 6, remainder 15) Concept / Approach Use simultaneous equations; substitute the division relation into the difference relation. Step-by-Step Calculation (6S + 15) − S = 13655S + 15 = 13655S = 1350 → S = 1350 ÷ 5 = 270 Verification L = 6S + 15 = 6(270) + 15 = 1620 + 15 = 1635Check difference: 1635 − 270 = 1365 ✓Check remainder: 1635 ÷ 270 = 6 remainder 15 ✓ Common Pitfalls Forgetting that the remainder must be smaller than the divisor (here 15 < 270). Final Answer 270

Question 4

Evaluate the square: 1397 × 1397.

Accepted Answer

Problem Restatement Calculate 13972 quickly without full long multiplication. Concept / Approach Use the identity (a − b)2 = a2 − 2ab + b2 with a = 1400 and b = 3. Step-by-Step Calculation 1397 = 1400 − 3(1400 − 3)2 = 14002 − 2 × 1400 × 3 + 32= 1,960,000 − 8,400 + 9 = 1,951,609 Verification Sanity check: 14002 = 1,960,000; the square should be slightly less — difference 8,391, consistent. Final Answer 1951609

Question 5

Evaluate: {(476 + 424)² − 4 × 476 × 424}.

Accepted Answer

Problem Restatement Compute {(476 + 424)2 − 4 × 476 × 424} using algebraic identities. Concept / Approach Recognize the identity: (a + b)2 − 4ab = a2 + 2ab + b2 − 4ab = a2 − 2ab + b2 = (a − b)2. Step-by-Step Calculation Let a = 476, b = 424(a − b)2 = (476 − 424)2 = 522 = 2704 Common Pitfalls Attempting brute-force squaring; spotting the identity is faster and reduces errors. Final Answer 2704

Question 6

If 64² − 36² = 20 × x, find the value of x.

Accepted Answer

Problem Restatement Use the difference-of-squares identity to evaluate 642 − 362 and solve for x in 20x. Concept / Approach (A2 − B2) = (A − B)(A + B). Step-by-Step Calculation 642 − 362 = (64 − 36)(64 + 36) = 28 × 100 = 28002800 = 20x → x = 2800 ÷ 20 = 140 Final Answer 140

Question 7

If n is a natural number, then the expression (6n² + 6n) is always divisible by which of the following?

Accepted Answer

Problem Restatement Find the highest guaranteed divisor among the choices for 6n2 + 6n when n is any natural number. Given / Factorization 6n2 + 6n = 6n(n + 1) Concept / Approach Among two consecutive integers n and (n + 1), one is even. Therefore n(n + 1) is even → divisible by 2. Since there is an explicit factor 6 = 2 × 3, the whole product has at least 2 × 2 × 3 = 12. Step-by-Step Reasoning 6n(n + 1) = (2 × 3) × n(n + 1)n(n + 1) contributes at least one factor 2Hence overall divisibility includes 2 × 2 × 3 = 12 Common Pitfalls Marking 6 as the answer; while true, 12 is also always a divisor and is larger. Assuming divisibility by 18; not guaranteed because an extra factor of 3 is not assured for all n. Final Answer 12

Question 8

Which of the following numbers will exactly divide (49¹⁵ − 1)?

Accepted Answer

Problem Restatement Test the divisibility of 4915 − 1 by the given options using number-theory facts. Concept / Approach For any integer a, an − 1 is always divisible by (a − 1). If n is even, an − 1 is also divisible by (a + 1); here n = 15 (odd), so this does not apply. Step-by-Step Reasoning a = 49, n = 15 → a − 1 = 48Thus 4915 − 1 is divisible by 48.Check a + 1 = 50: 49 ≡ −1 (mod 50) → 4915 − 1 ≡ (−1)15 − 1 = −2 ≠ 0 (mod 50) Final Answer 48

Question 9

Find the unit digit of the product: 784 × 618 × 917 × 463.

Accepted Answer

Problem Restatement Determine only the units digit (last digit) of the large product 784 × 618 × 917 × 463. Concept / Approach The unit digit of a product depends only on the unit digits of the factors. Step-by-Step Calculation Consider unit digits: 784 (→ 4), 618 (→ 8), 917 (→ 7), 463 (→ 3)Multiply sequentially (keep only unit digit at each step):4 × 8 = 32 → unit digit 22 × 7 = 14 → unit digit 44 × 3 = 12 → unit digit 2 Common Pitfalls Unnecessary full multiplication; tracking only unit digits suffices. Final Answer 2

Question 10

Which natural number nearest to 9217 is exactly divisible by 88?

Accepted Answer

Problem Restatement Find the closest multiple of 88 to 9217. Concept / Approach Compute the nearest multiples by division and compare distances. Step-by-Step Calculation Compute 9217 ÷ 88.88 × 104 = 9152 (since 88 × 100 = 8800; 8800 + 352 = 9152)Next multiple: 88 × 105 = 9240Distances: |9217 − 9152| = 65; |9240 − 9217| = 23Nearest multiple is 9240 Quick Check 9216 is not a multiple of 88 (88 × 104 = 9152; +64 would be 9216, but 64 is not 88). Final Answer 9240

Question 11

Evaluate: 107 × 107 + 93 × 93. What is the value?

Accepted Answer

Given data Compute 1072 + 932. Concept / Approach Square each term separately and add; or use identities to cross-check. Useful identity for verification: (a + b)2 = a2 + 2ab + b2 ⇒ a2 + b2 = (a + b)2 − 2ab. Step-by-step calculation 1072 = 11,449932 = 8,649Sum = 11,449 + 8,649 = 20,098 Verification / Alternative (107 + 93)2 − 2·107·93 = 2002 − 2·9,951 = 40,000 − 19,902 = 20,098 Common pitfalls Arithmetic slip while squaring near round numbers; always verify with an identity. Final Answer 20098

Question 12

A number gives quotient 269 and remainder 0 when divided by 68. When the same number is divided by 67, what is the remainder?

Accepted Answer

Given data N = 68 × 269 (since remainder is 0 on division by 68). Find N mod 67. Concept / Approach Use modular arithmetic: (ab) mod m = [(a mod m)(b mod m)] mod m. Since 68 ≡ 1 (mod 67), N ≡ 1 × (269 mod 67) (mod 67). Step-by-step calculation 269 = 67 × 4 + 1 ⇒ 269 mod 67 = 1Therefore, N mod 67 = 1 Verification N = 68 × 269 = 18,292; 18,292 ÷ 67 = 272 remainder 1. Common pitfalls Reducing both factors needlessly; recognizing 68 ≡ 1 (mod 67) simplifies immediately. Final Answer 1

Question 13

What least number must be added to 1056 so that the sum is exactly divisible by 23?

Accepted Answer

Given data Target: smallest k ≥ 0 with (1056 + k) divisible by 23. Concept / Approach Compute 1056 mod 23; add the complement to the next multiple. Step-by-step calculation 23 × 45 = 1,035Remainder r = 1,056 − 1,035 = 21Need k = 23 − r = 2 ⇒ 1,056 + 2 = 1,058 = 23 × 46Thus, k = 2 Common pitfalls Choosing remainder itself instead of the complement to reach the next multiple. Final Answer 2

Question 14

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

Accepted Answer

Given data Options: 1537, 4096, 6241, 7744. Concept / Approach Squares in base-10 cannot end with 2, 3, 7, or 8. Recognize known perfect squares to confirm. Step-by-step reasoning 4096 = 6426241 = 7927744 = 8821537 ends with 7 ⇒ cannot be a perfect square Common pitfalls Assuming any odd ending can be a square; endings 2, 3, 7, 8 are impossible for squares. Final Answer 1537

Question 15

Which of the following is a prime number?

Accepted Answer

Given data Options: 31, 51, 91, 121. Concept / Approach Prime test up to √n. Checks 31: √31 ≈ 5.57; not divisible by 2, 3, 5 ⇒ prime51 = 3 × 17 ⇒ composite91 = 7 × 13 ⇒ composite121 = 11 × 11 ⇒ composite Common pitfalls Forgetting 91 = 7 × 13 (often mistaken for prime). Final Answer 31

Question 16

What is the sum of the first five prime numbers?

Accepted Answer

Given data The first five primes are 2, 3, 5, 7, 11. Concept / Approach Primes are natural numbers greater than 1 with exactly two distinct positive divisors. Calculation Sum = 2 + 3 + 5 + 7 + 11 = 28 Common pitfalls Including 1 as prime (it is not prime). Final Answer 28

Question 17

Find the sum of all even natural numbers between 1 and 31.

Accepted Answer

Given data Even numbers from 2 to 30 inclusive form an arithmetic progression (AP). Concept / Approach AP sum: S = n/2 (first + last); here, difference d = 2. Step-by-step calculation First term a = 2; last term l = 30; d = 2Number of terms n = ((l − a)/d) + 1 = ((30 − 2)/2) + 1 = 14 + 1 = 15Sum S = n/2 × (a + l) = 15/2 × 32 = 15 × 16 = 240 Verification Pairwise: (2+30), (4+28), … each = 32; 15/2 pairs ⇒ 15 × 16 = 240. Common pitfalls Accidentally including 32 or excluding 2. Final Answer 240

Question 18

How many prime numbers are less than 50?

Accepted Answer

Given data List primes below 50 and count them. Concept / Approach Primes below 50 can be enumerated or found by sieving. Enumeration 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47Count = 15 Common pitfalls Including 1 or 49 (72) by mistake. Final Answer 15

Question 19

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

Accepted Answer

Given data Consecutive integers from 51 to 100 inclusive. Concept / Approach Arithmetic series: S = n/2 × (first + last) or S = n × average. Step-by-step calculation Number of terms n = 100 − 51 + 1 = 50Average = (51 + 100)/2 = 151/2 = 75.5Sum S = n × average = 50 × 75.5 = 3,775 Verification / Alternative Pairing method: (51+100), (52+99), … each pair = 151; 25 pairs ⇒ 25 × 151 = 3,775. Common pitfalls Forgetting to include both endpoints 51 and 100. Final Answer 3775

Question 20

How many of the following numbers are divisible by 132? 264, 396, 462, 792, 968, 2178, 5184, 6336

Accepted Answer

Given data Test divisibility by 132 for: 264, 396, 462, 792, 968, 2178, 5184, 6336. Concept / Approach Prime factorize 132: 132 = 22 × 3 × 11. Hence, a number must be divisible by 4, by 3, and by 11 to be divisible by 132. Quick checks: /4 ⇒ last two digits multiple of 4; /3 ⇒ sum of digits multiple of 3; /11 ⇒ alternating sum rule. Step-by-step checks 264: 132 × 2 ⇒ divisible ✓ 396: 132 × 3 ⇒ divisible ✓ 462: /4? last two digits 62 (not /4) ⇒ not divisible ✗ 792: 132 × 6 ⇒ divisible ✓ 968: /4 yes (68/4), /3? 9+6+8=23 (no) ⇒ not divisible ✗ 2178: /3 yes (2+1+7+8=18), /4? 78/4 not integer ⇒ not divisible ✗ 5184: /4 yes (84/4), /3 yes (5+1+8+4=18), /11? (5+8) − (1+4) = 8 ≠ 0, ±11 ⇒ not divisible ✗ 6336: 132 × 48 ⇒ divisible ✓ Count Divisible: 264, 396, 792, 6336 ⇒ total = 4 Common pitfalls Checking only /4 and /3 but forgetting the /11 rule for divisibility by 132. Final Answer 4

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