The H.C.F. and L.C.M. of two numbers are 13 and 1989, respectively. If one number is 117, determine the other number.

Introduction / Context:
We use the product identity for two integers a and b: a*b = gcd(a, b) * lcm(a, b). Given gcd and lcm along with one number, the other number follows immediately from algebraic manipulation.

Given Data / Assumptions:

• gcd = 13.
• lcm = 1989.
• One number a = 117.
• Find b.

Concept / Approach:
Compute b = (gcd * lcm) / a. Simplify before multiplying if helpful, to avoid large intermediate values and reduce arithmetic errors.

Step-by-Step Solution:

Verification / Alternative check:
Check gcd(117, 221) = 13 and lcm(117, 221) = (117*221)/13 = 1989. Both conditions are satisfied, verifying the result.

Why Other Options Are Wrong:

• 121, 131, 231, 143: These candidates do not simultaneously satisfy the given gcd and lcm with 117 under the product identity.

Common Pitfalls:

• Cancelling incorrectly or multiplying before simplifying, which can lead to arithmetic slips.

Final Answer:
221