Question 1

For two consecutive even integers, what is the largest integer that always divides the difference of their squares?

Accepted Answer

Given data Let the consecutive even integers be 2n and 2n + 2. Concept / Approach Use the identity a^2 − b^2 = (a − b)(a + b) and simplify for consecutive even numbers. Step-by-step calculation (2n + 2)^2 − (2n)^2 = [(2n + 2) − 2n] · [(2n + 2) + 2n]= 2 · (4n + 2) = 8n + 4 = 4(2n + 1) Verification / Notes 2n + 1 is odd for all integers n, so 8n + 4 is exactly 4 times an odd number. Hence the result is always divisible by 4; not guaranteed by 8 (needs an extra factor 2 in 2n + 1, which never happens). Common pitfalls Assuming divisibility by 8; 2n + 1 is never even. Final Answer Always divisible by 4.

Question 2

A number leaves remainder 11 when divided by 13 and remainder 9 when divided by 17. Find the least such number.

Accepted Answer

Given data N ≡ 11 (mod 13) N ≡ 9 (mod 17) Concept / Approach Chinese Remainder Theorem on mod 13 and mod 17 (coprime, so unique solution mod 221). Step-by-step calculation Let N = 11 + 13k.Impose mod 17: 11 + 13k ≡ 9 (mod 17) ⇒ 13k ≡ −2 ≡ 15 (mod 17).Since 13·4 ≡ 52 ≡ 1 (mod 17), multiply both sides by 4 ⇒ k ≡ 15·4 ≡ 60 ≡ 9 (mod 17).Smallest k = 9 ⇒ N = 11 + 13·9 = 128.General solution: N = 128 + 221t, t ∈ ℤ. Verification 128 ÷ 13 = 9 remainder 11 ✓ 128 ÷ 17 = 7 remainder 9 ✓ Final Answer Least number = 128.

Question 3

Find the largest 5-digit number that is exactly divisible by 91.

Accepted Answer

Given data We need the greatest multiple of 91 that does not exceed 99,999. Concept / Approach Compute ⌊99999 ÷ 91⌋ and multiply back by 91. Step-by-step calculation 91 × 1100 = 100,100 (too large)So try 1100 − 2 = 1098 ⇒ 91 × 1098 = 100,100 − 182 = 99,918 Verification 99,918 + 91 = 100,009 > 99,999 ⇒ 99,918 is indeed the largest 5-digit multiple. Final Answer Largest 5-digit multiple of 91 is 99,918.

Question 4

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

Accepted Answer

Given data Even numbers between 1 and 31: 2, 4, 6, …, 30 (an AP with d = 2). Concept / Approach Sum of AP: S = n/2 · (first + last). Step-by-step calculation a = 2, l = 30, d = 2n = ((l − a)/d) + 1 = ((30 − 2)/2) + 1 = 15S = 15/2 · (2 + 30) = 15/2 · 32 = 15 · 16 = 240 Common pitfalls Including 32 or excluding 2—stick to 2 through 30 inclusive. Final Answer Sum = 240.

Question 5

On multiplying a number by 7, the product is a number consisting only of the digit 3. Find the smallest such product.

Accepted Answer

Given data We need the smallest repdigit of 3s (say with k digits) that is divisible by 7. Concept / Approach Let ak be the k-digit number of all 3s. Recurrence (mod 7): ak+1 ≡ 10·ak + 3 (mod 7) and since 10 ≡ 3 (mod 7), ak+1 ≡ 3(ak + 1) (mod 7). Step-by-step residues (mod 7) a1 = 3 ≡ 3a2 ≡ 3(3 + 1) = 12 ≡ 5a3 ≡ 3(5 + 1) = 18 ≡ 4a4 ≡ 3(4 + 1) = 15 ≡ 1a5 ≡ 3(1 + 1) = 6 ≡ 6a6 ≡ 3(6 + 1) = 21 ≡ 0 ⇒ divisible by 7 Conclusion The smallest all-3s multiple of 7 has 6 digits: 333333. Final Answer Smallest such product = 333333.

Question 6

A number leaves remainder 75 when divided by 296. What is the remainder when the same number is divided by 37?

Accepted Answer

Given data N ≡ 75 (mod 296). Note: 296 = 8 × 37. Concept / Approach Reduce the remainder under the smaller modulus: N mod 37 = 75 mod 37 since 296 is a multiple of 37. Step-by-step calculation 75 ÷ 37 = 2 remainder 1.Therefore, N ≡ 1 (mod 37). Final Answer Remainder = 1.

Question 7

A number leaves remainders 1 and then 4 when divided successively by 4 and 5. If instead it is divided successively by 5 and then 4, what are the respective remainders?

Accepted Answer

Given data Successive division by 4 then 5 yields remainders: 1 (mod 4), then 4 (mod 5). Find the remainders if the order is reversed: first divide by 5, then divide the quotient by 4. Concept / Approach Translate successive remainders into algebra and back substitute. Then recompute remainders when the order of the divisors is reversed. Step-by-step calculation Let the original number be N.Since division by 4 leaves remainder 1: N = 4a + 1 for some integer a.Now divide the quotient a by 5 and get remainder 4: a = 5b + 4 for some integer b.Therefore N = 4(5b + 4) + 1 = 20b + 17. Reversed order: first by 5, then by 4 First remainder (mod 5): N mod 5 = (20b + 17) mod 5 = 0 + 2 = 2.Quotient after dividing by 5: Q = (N − 2)/5 = (20b + 15)/5 = 4b + 3.Second remainder (mod 4): Q mod 4 = (4b + 3) mod 4 = 3. Verification / Alternative Choose b = 0 (the least case): N = 20·0 + 17 = 17.Check original order: 17 ÷ 4 ⇒ q = 4, r = 1; then 4 ÷ 5 ⇒ q = 0, r = 4 ✓Check reversed order: 17 ÷ 5 ⇒ q = 3, r = 2; then 3 ÷ 4 ⇒ q = 0, r = 3 ✓ Common pitfalls Confusing simultaneous congruences with successive division. Here the second remainder applies to the quotient from the first division. Trying to combine the divisors into a single modulus; that does not model successive operations correctly. Final Answer On reversing the order, the remainders are 2 and 3.

Question 8

Evaluate: 935421 × 625.

Accepted Answer

Given data Compute 935,421 × 625. Concept / Approach Use 625 = 10,000 ÷ 16 to avoid long multiplication. Step-by-step calculation 935,421 × 10,000 = 9,354,210,000Now divide by 16: 9,354,210,000 ÷ 16 = 584,638,125 Verification 584,638,125 × 16 = 9,354,210,000 (exact), then ×(1/16) × 10,000 = ×625. Final Answer 584638125

Question 9

Evaluate: 753 × 753 + 247 × 247 − 753 × 247.

Accepted Answer

Given data a = 753, b = 247, with a + b = 1000. Concept / Approach Rewrite a^2 + b^2 − ab via (a + b)^2: a^2 + b^2 = (a + b)^2 − 2ab ⇒ expression = (a + b)^2 − 3ab. Step-by-step calculation (a + b)^2 = 1000^2 = 1,000,000ab = 753 × 247 = 753 × (250 − 3) = 188,250 − 2,259 = 185,991Therefore, value = 1,000,000 − 3 × 185,991 = 1,000,000 − 557,973 = 442,027 Verification / Alternative Direct expansion a^2 + b^2 − ab matches the algebraic transformation result. Final Answer 442027

Question 10

A number is divided successively by 4, then 5, then 6 and the remainders are 2, 3 and 4 respectively. Find the least such number.

Accepted Answer

Given data Successive division remainders: 2 (by 4), then 3 (by 5), then 4 (by 6). Concept / Approach Back-substitute from the last quotient to construct N explicitly. Step-by-step calculation Let after first division by 4: N = 4a + 2.Then a ÷ 5 leaves remainder 3 ⇒ a = 5b + 3.Then b ÷ 6 leaves remainder 4 ⇒ b = 6c + 4.Thus N = 4(5(6c + 4) + 3) + 2 = 4(30c + 23) + 2 = 120c + 94.Least positive (c = 0): N = 94. Verification 94 ÷ 4 ⇒ q = 23, r = 2 23 ÷ 5 ⇒ q = 4, r = 3 4 ÷ 6 ⇒ q = 0, r = 4 Final Answer Least number = 94.

Question 11

Which of the following is not a prime number?

Accepted Answer

Given data Options: 97, 121, 83, 89 Concept / Approach A prime has exactly two distinct positive divisors: 1 and itself. 1221? No; check each option up to its square root. For a perfect square n = p^2 with p > 1, n is never prime. Step-by-step checks 121 = 11 × 11 → composite97, 83, 89: not divisible by any prime ≤ 10 (2, 3, 5, 7) → each is prime Common pitfalls Confusing a large odd number with being prime. Always test by small primes up to √n. Final Answer The non-prime is 121.

Question 12

If the number 918762 is divisible by 8, what is the smallest digit that can replace ?

Accepted Answer

Given data Number: 918762 Divisibility condition: divisible by 8 Concept / Approach A number is divisible by 8 if its last three digits form a number divisible by 8. Step-by-step calculation Last three digits = 62 → consider 602, 612, 622, 632, 642, 652, 662, 672, 682, 692Check divisibility by 8: 632 ÷ 8 = 79 (exact)Smallest digit that works is 3 Verification Other smaller candidates fail: 602 mod 8 = 2, 612 mod 8 = 4, 622 mod 8 = 6. Common pitfalls Testing the full number unnecessarily; only the last three digits matter for divisibility by 8. Final Answer Smallest possible digit is 3.

Question 13

Evaluate: (12)³ × 64 ÷ 432

Accepted Answer

Given data Expression: (12)3 × 64 ÷ 432 Concept / Approach Simplify using factor cancellation before multiplying to avoid large numbers. Step-by-step calculation 123 = 12 × 12 × 12 = 1728(1728 × 64) ÷ 432 = (1728 ÷ 432) × 641728 ÷ 432 = 4Therefore value = 4 × 64 = 256 Verification / Alternative 432 = 27 × 16; also 64 ÷ 16 = 4; expression = 123 × 4 ÷ 27 = 1728 × 4 ÷ 27 = 6912 ÷ 27 = 256. Common pitfalls Multiplying first then dividing can invite arithmetic errors. Cancel common factors early. Final Answer 256

Question 14

Which of the following numbers is divisible by 3, 7, 9, and 11?

Accepted Answer

Given data Divisors: 3, 7, 9, 11 Options: 1980, 2310, 2079, 5005 Concept / Approach LCM(3, 7, 9, 11) = 9 × 7 × 11 = 693. The number must be a multiple of 693. Step-by-step checks 693 × 3 = 2079 (candidate)1980: not divisible by 9 (sum of digits 1+9+8+0=18 yes) but 1980 ÷ 11 is not integer2310: divisible by 3 and 7 and 11, but not by 9 (sum 2+3+1+0=6)5005: divisible by 5, 7, 11 but not by 9 (sum 10) Final Answer Only 2079 is divisible by all 3, 7, 9, and 11.

Question 15

Find the value of ?: ? − 19657 − 33994 = 9999

Accepted Answer

Given data Equation: ? − 19657 − 33994 = 9999 Concept / Approach Rearrange to isolate the unknown. Step-by-step calculation ? = 9999 + 19657 + 339949999 + 19657 = 2965629656 + 33994 = 63650 Verification Check: 63650 − 19657 − 33994 = 63650 − 53651 = 9999 ✓ Final Answer ? = 63650.

Question 16

If 653xy is divisible by 80, find x + y

Accepted Answer

Given data Number: 653xy Divisibility requirement: by 80 Concept / Approach 80 = 16 × 5 (coprime). So the number must be divisible by 5 and by 16. Divisible by 5 → last digit y = 0. Divisible by 16 → last four digits are divisible by 16. Step-by-step calculation With y = 0, last four digits are 53x0 = 5300 + 10x.Compute (5300 + 10x) mod 16.5300 mod 16 = 4 (since 16 × 331 = 5296)We need 4 + 10x ≡ 0 (mod 16).Reduce: 10x ≡ 12 (mod 16) ⇒ divide by 2 ⇒ 5x ≡ 6 (mod 8).Inverse of 5 mod 8 is 5 (since 5 × 5 ≡ 1 mod 8).x ≡ 6 × 5 ≡ 30 ≡ 6 (mod 8). As a digit, x = 6.Thus y = 0 and x + y = 6. Verification Last four digits 5360 ÷ 16 = 335 (exact); last digit 0 satisfies divisibility by 5. Common pitfalls Testing divisibility by 8 instead of 16; 80 requires 16 × 5, not 8 × 10. Final Answer x + y = 6.

Question 17

If 481*673 is divisible by 9, find the smallest digit that can replace

Accepted Answer

Given data Number: 481673 Divisibility requirement: by 9 Concept / Approach A number is divisible by 9 if the sum of its digits is a multiple of 9. Step-by-step calculation Sum of known digits = 4 + 8 + 1 + 6 + 7 + 3 = 29We need 29 + * ≡ 0 (mod 9) ⇒ 29 ≡ 2 (mod 9), so * ≡ 7 (mod 9)Smallest digit satisfying this is 7 Verification 29 + 7 = 36, a multiple of 9. Common pitfalls Using divisibility by 3 instead of 9; both use digit sums but mod 9 is required here. Final Answer Smallest digit is 7.

Question 18

Evaluate: 397 × 397 + 104 × 104 + 2 × 397 × 104

Accepted Answer

Given data Expression: a2 + b2 + 2ab with a = 397, b = 104 Concept / Approach Use (a + b)2 = a2 + 2ab + b2. Step-by-step calculation a + b = 397 + 104 = 501Value = (a + b)2 = 50125012 = (500 + 1)2 = 5002 + 2 × 500 × 1 + 1 = 250000 + 1000 + 1 = 251001 Verification Direct multiplication yields the same result; the identity avoids heavy arithmetic. Final Answer 251001

Question 19

If 42573* is divisible by 72, find the minimum value of

Accepted Answer

Given data Number: 42573 Divisibility requirement: by 72 Concept / Approach Since 72 = 8 × 9 and gcd(8, 9) = 1, the number must be divisible by 8 and by 9 simultaneously. Step-by-step calculation: divisibility by 8 Only the last three digits matter: 73*Nearest multiples of 8 around 730 are 728 and 736 and 744736 fits the pattern 73* ⇒ * = 6 Step-by-step calculation: divisibility by 9 Sum of digits (without *) = 4 + 2 + 5 + 7 + 3 = 21We need (21 + *) to be a multiple of 9 ⇒ * ≡ 6 (mod 9)Smallest digit satisfying both tests is 6 Verification 425736 ÷ 8 = 53217; 4 + 2 + 5 + 7 + 3 + 6 = 27 divisible by 9. Common pitfalls Checking only one of the two conditions; 72 requires both 8 and 9. Final Answer Minimum value of * is 6.

Question 20

Evaluate: 1904 × 1904

Accepted Answer

Given data Compute 19042. Concept / Approach Use (a + b)2 with a = 1900, b = 4 to expand quickly. Step-by-step calculation (1900 + 4)2 = 19002 + 2 × 1900 × 4 + 4219002 = 3,610,0002 × 1900 × 4 = 15,20042 = 16Sum = 3,610,000 + 15,200 + 16 = 3,625,216 Verification / Alternative Compute 1904 × 1904 by standard multiplication; the result matches 3,625,216. Common pitfalls Dropping the middle term 2ab in the square expansion. Final Answer 3625216

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