If a and b are two co-prime natural numbers (with a greater than b), what is the least common multiple (LCM) of a and b?

Introduction:
This conceptual question tests your understanding of co-prime numbers and their least common multiple (LCM). Co-prime numbers are pairs of integers whose highest common factor (HCF) is 1, even though each number may be greater than 1.

Given Data / Assumptions:

• a and b are natural numbers.
• a and b are co-prime, so HCF(a, b) = 1.
• a is greater than b.
• We need the LCM of a and b.

Concept / Approach:
For any two positive integers a and b, there is a fundamental relationship: a × b = HCF(a, b) × LCM(a, b). If a and b are co-prime, their HCF is 1. Substituting this into the formula directly gives the LCM of two co-prime numbers.

Step-by-Step Solution:
Start with the general formula: a × b = HCF(a, b) × LCM(a, b) Given that a and b are co-prime, HCF(a, b) = 1. So the formula becomes: a × b = 1 × LCM(a, b) Therefore, LCM(a, b) = a × b Thus, for co-prime numbers, the LCM is simply the product ab

Verification / Alternative check:
Take a simple example: a = 3 and b = 4. These numbers are co-prime. The multiples of 3 are 3, 6, 9, 12, 15, and so on. The multiples of 4 are 4, 8, 12, 16, and so on. The smallest common multiple is 12, which is 3 × 4. This matches the rule LCM = ab for co-prime numbers.

Why Other Options Are Wrong:
a and b by themselves cannot be the LCM because the LCM must be a number that is a multiple of both a and b. a + b is not guaranteed to be divisible by either a or b, so it is not a general formula for the LCM of co-prime numbers. Only ab always works.

Common Pitfalls:
Learners sometimes think that the LCM of co-prime numbers is their sum or the larger of the two numbers. This is incorrect. The product ab is the smallest number that has all the prime factors of both a and b with correct multiplicities when they share no common prime factor.

Final Answer:
For two co-prime natural numbers a and b, the least common multiple is ab.