Two numbers are coprime and their product is 117. What is their L.C.M. (least common multiple)?

Introduction / Context:
The relationship between the GCD (HCF) and LCM of two integers a and b is a*b = gcd(a, b) * lcm(a, b). If a and b are coprime, their gcd is 1, which simplifies the relationship greatly.

Given Data / Assumptions:

• a and b are coprime ⇒ gcd(a, b) = 1.
• Product a*b = 117.

Concept / Approach:
Using a*b = gcd(a, b) * lcm(a, b), substitute gcd(a, b) = 1 and a*b = 117 to obtain lcm(a, b) directly. No factorization of 117 is required for the conclusion, although 117 = 9*13 shows a possible coprime pair is not necessary here.

Step-by-Step Solution:

Verification / Alternative check:
If a = 9 and b = 13 (which are coprime), lcm(9, 13) = 117 indeed, matching the formula. Any coprime pair with product 117 yields the same LCM value 117.

Why Other Options Are Wrong:

• 9, 13, 39, 81: None equals the product 117 for coprime numbers; only 117 satisfies the identity for all coprime factorizations of 117.

Common Pitfalls:

• Assuming the LCM is one of the factors instead of using the product–gcd–lcm identity.

Final Answer:
117