Find the greatest number that divides 1657 and 2037 leaving remainders 6 and 5 respectively.

Given data

• 1657 leaves remainder 6; 2037 leaves remainder 5 when divided by the required number.

Concept / Approach

• If numbers leave (possibly different) remainders r₁, r₂ on division by d, then d divides the difference of the adjusted numbers: gcd(1657 − 6, 2037 − 5).

Step-by-step gcd

1657 − 6 = 16512037 − 5 = 2032gcd(1651, 2032):2032 − 1651 = 3811651 ÷ 381 = 4 remainder 127381 ÷ 127 = 3 remainder 0 ⇒ gcd = 127

Verification

1657 mod 127 = 6 and 2037 mod 127 = 5.

Common pitfalls

• Taking gcd of 1657 and 2037 directly without adjusting for remainders.

Final Answer

Greatest number = 127.