The greatest number that divides 1657 and 2037 leaving remainders 6 and 5 respectively is:

Take away the remainders so the division becomes exact: 1657 − 6 = 1651 and 2037 − 5 = 2032. The required number is HCF(1651, 2032). Since $1651=127\times 13$ and $2032=127\times 16$, and 13 and 16 share nothing, the HCF is 127 (comfortably larger than both remainders). Answering 381 confuses the HCF with the difference 2032 − 1651, and 13 is only the co-factor.