The greatest number that leaves remainders 6 and 5 when dividing 1657 and 2037 respectively is equal to what value?

Introduction / Context:
This is a classic highest common factor (H.C.F.) question where a divisor leaves different remainders when dividing two numbers. It tests whether you understand how to translate remainder information into an equivalent problem involving differences and greatest common divisors, a concept that appears very often in aptitude and number theory questions.

Given Data / Assumptions:

• When the required number divides 1657, the remainder is 6.
• When the same number divides 2037, the remainder is 5.
• The number we seek is the greatest possible integer that satisfies both conditions.

Concept / Approach:
If a number d divides 1657 and leaves remainder 6, then 1657 - 6 is exactly divisible by d. Similarly, if the same number leaves remainder 5 on dividing 2037, then 2037 - 5 is divisible by d. Therefore, d must divide the difference between these adjusted values as well. So we first compute 1651 and 2032, then find the H.C.F. of these two numbers to get the greatest possible divisor satisfying both remainder conditions.

Step-by-Step Solution:
Step 1: Let the required greatest number be d.Step 2: From 1657 with remainder 6, we get 1657 - 6 = 1651, which is divisible by d.Step 3: From 2037 with remainder 5, we get 2037 - 5 = 2032, which is also divisible by d.Step 4: Therefore, d is a common divisor of 1651 and 2032.Step 5: The greatest such divisor is the H.C.F. of 1651 and 2032.Step 6: Find H.C.F.: 2032 - 1651 = 381.Step 7: Next, 1651 ÷ 381 gives a remainder: 381 * 4 = 1524, 1651 - 1524 = 127.Step 8: Now compute 381 ÷ 127. This is exact because 127 * 3 = 381.Step 9: Hence, the H.C.F. of 1651 and 2032 is 127, so the required greatest number is 127.

Verification / Alternative check:
Check 1657 ÷ 127: 127 * 13 = 1651 and 1657 - 1651 = 6, correct remainder.Check 2037 ÷ 127: 127 * 16 = 2032 and 2037 - 2032 = 5, correct remainder. As both conditions are satisfied, 127 is confirmed as the correct greatest number.

Why Other Options Are Wrong:
Option a (123), option c (235), and option d (305) do not simultaneously satisfy the remainder conditions for both 1657 and 2037. If you perform the division and check the remainders, you will see that they either fail for one of the numbers or do not divide the adjusted values 1651 and 2032 exactly. Hence, they cannot be the required greatest number.

Common Pitfalls:
Students sometimes attempt to find a number that directly divides the original numbers with the given remainders without converting to the equivalent H.C.F. problem. Another frequent error is to subtract the remainders incorrectly or to compute the H.C.F. carelessly. Always remember that you must work with 1657 - 6 and 2037 - 5, not with 1657 and 2037 directly, when the remainders are known.

Final Answer:
The greatest number that satisfies the given conditions is 127.