The product of two co-prime numbers is 117. What should be the L.C.M. of these two numbers?

Introduction / Context:
This is a concept-based question on co-prime numbers and the relationship between highest common factor (H.C.F.) and least common multiple (L.C.M.). It tests your theoretical understanding rather than long calculation, making it a quick scoring problem if you know the basic rule.

Given Data / Assumptions:

• The two numbers are co-prime, meaning their H.C.F. is 1.
• The product of the two numbers is 117.
• We are asked to find the L.C.M. of the two numbers.

Concept / Approach:
For any two positive integers a and b, the relation between their product, H.C.F. (denoted h), and L.C.M. (denoted L) is: a * b = h * L. When a and b are co-prime, their H.C.F. is 1. That means a * b = 1 * L, or L = a * b. So for co-prime numbers, their L.C.M. is simply equal to the product of the numbers.

Step-by-Step Solution:
Step 1: Let the two co-prime numbers be a and b.Step 2: Given that a and b are co-prime, H.C.F.(a, b) = 1.Step 3: The product of the numbers is a * b = 117.Step 4: Use the formula a * b = H.C.F. * L.C.M.Step 5: Substitute H.C.F. = 1, so 117 = 1 * L.C.M.Step 6: Therefore L.C.M. = 117.

Verification / Alternative check:
Since a and b are co-prime, any factorization of 117 into primes can represent the two numbers by grouping the prime factors in any way. For example, 117 = 9 * 13, where gcd(9, 13) = 1. The L.C.M. of 9 and 13 is (9 * 13) / 1 = 117, which matches our result. Any other co-prime factor pair will still produce an L.C.M. equal to 117.

Why Other Options Are Wrong:
Option a (1) is incorrect because it confuses H.C.F. with L.C.M.; co-prime numbers have H.C.F. 1, not L.C.M. 1. Option c (Equal to their H.C.F) is also wrong because their H.C.F. is 1, while the L.C.M. here is clearly larger. Option d (cannot be calculated) is incorrect, as there is enough information to find the L.C.M. using the standard formula.

Common Pitfalls:
Students sometimes mix up H.C.F. and L.C.M., assuming that if numbers are co-prime, both must be small. Others forget the formula a * b = H.C.F. * L.C.M. Knowing this relationship is crucial to quickly answering many exam questions without trial and error.

Final Answer:
The L.C.M. of the two co-prime numbers is 117.