Step-by-step Solution:

To find the greatest number that divides 1657 and 2037, leaving remainders 6 and 5 respectively, subtract the remainders:

• 1657 - 6 = 1651
• 2037 - 5 = 2032

The greatest number is the HCF of 1651 and 2032. Using the Euclidean algorithm:

• 2032 = 1651 × 1 + 381
• 1651 = 381 × 4 + 127
• 381 = 127 × 3 + 0

HCF is 127.

Answer: (A) 127