Find the greatest divisor from non-equal remainders The greatest number that divides 1657 and 2037 leaving remainders 6 and 5 respectively is:

Introduction / Context:
When a number d divides two numbers leaving remainders r1 and r2, it divides the corresponding differences (N1 - r1) and (N2 - r2) exactly. The required greatest divisor is then gcd(N1 - r1, N2 - r2).

Given Data / Assumptions:

• N1 = 1657 leaves remainder 6.
• N2 = 2037 leaves remainder 5.
• d | (N1 - 6) and d | (N2 - 5).

Concept / Approach:
Compute A = 1657 - 6 and B = 2037 - 5, then find gcd(A, B) using the Euclidean algorithm.

Step-by-Step Solution:
A = 1651, B = 2032.B - A = 381.Compute gcd(1651, 2032): gcd(1651, 381) since 2032 - 1651 = 381.1651 = 381 * 4 + 127.381 = 127 * 3 + 0 ⇒ gcd = 127.

Verification / Alternative check:
Check divisibility: 1651 ÷ 127 = 13; 2032 ÷ 127 = 16; hence 127 divides both differences and works.

Why Other Options Are Wrong:
235, 260, 305, 381 do not evenly divide both 1651 and 2032, or are not the greatest common divisor.

Common Pitfalls:
Taking gcd of 1657 and 2037 directly (ignoring remainders), or subtracting the wrong remainders.

Final Answer:
127