Find the greatest number which on dividing 1661 and 2045, leaves a remainder…

Method: If a number divides 1661 and 2045 leaving remainders 10 and 13 respectively, then it divides the differences 1661 - 10 and 2045 - 13.

Compute the adjusted numbers:

• 1661 - 10 = 1651

• 2045 - 13 = 2032

The required greatest number is gcd(1651, 2032). Use the Euclidean algorithm:

• 2032 = 1651 × 1 + 381

• 1651 = 381 × 4 + 127

• 381 = 127 × 3 + 0

So gcd(1651, 2032) = 127.

Answer: 127

Quick check: 1651 = 127 × 13 and 2032 = 127 × 16, so 127 divides both adjusted numbers and is the greatest such divisor.