If a and b are two co prime positive integers with a greater than b, what is the least common multiple (LCM) of a and b expressed in terms of a and b?

Introduction / Context:
This conceptual question focuses on the properties of co prime numbers. Two numbers are co prime if their highest common factor (HCF) is 1. The relationship between the product of two numbers, their HCF, and their least common multiple (LCM) simplifies significantly when the numbers are co prime. Understanding this is important for solving many number theory and arithmetic problems quickly.

Given Data / Assumptions:

• a and b are positive integers.
• a and b are co prime, which means HCF(a, b) = 1.
• a is greater than b.
• We need the LCM of a and b in algebraic form.

Concept / Approach:
For any two positive integers a and b, the fundamental identity is: a * b = HCF(a, b) * LCM(a, b). If a and b are co prime, then HCF(a, b) = 1. Substituting this into the identity simplifies the relationship, making LCM(a, b) equal to the product a * b.

Step-by-Step Solution:
Given HCF(a, b) = 1 because a and b are co prime. Using the identity: a * b = HCF(a, b) * LCM(a, b). This becomes a * b = 1 * LCM(a, b). Therefore, LCM(a, b) = a * b. So the LCM of two co prime numbers a and b is ab.

Verification / Alternative check:
Consider simple co prime examples: Example 1: a = 3, b = 4. LCM(3, 4) = 12, and ab = 3 * 4 = 12. Example 2: a = 5, b = 7. LCM(5, 7) = 35, and ab = 5 * 7 = 35. In each case, the LCM equals the product of the numbers, confirming the general result for co prime pairs.

Why Other Options Are Wrong:
a or b alone cannot be the LCM because the LCM must be divisible by both numbers, and a single number will not be divisible by the other unless they are equal, which need not be true. a + b and a - b generally do not satisfy the divisibility conditions required for the LCM, except in special accidental cases. They are not guaranteed formulas for all co prime pairs.

Common Pitfalls:
Some learners confuse co prime numbers with consecutive numbers or with numbers having a small HCF. Others may forget the product identity and try to guess based on the size of expressions. Remember that co prime means HCF equals 1, and this makes the LCM equal to the plain product a * b by the standard identity.

Final Answer:
ab