Question

Is the statement below always, sometimes, or never true? Give at least two examples to support your reasoning. The LCM of the two numbers is the product of the two numbers.

Answer 1

The statement is sometimes true.

Explanation:

The LCM of two numbers is the product of the two numbers when they are coprime or relatively prime. Coprimes are numbers whose only common factor is 1. Examples are:

1. 4 and 7
2. 5 and 23
3. 48 and 49

Note that the numbers do not have to be prime numbers as the third example indicates. Generally, a prime number is coprime with any other number.

When there is a common factor other than 1, then the LCM of the two numbers is not their product. This is because their product will be divisible by the common factor, contradicting the fact it is the LCM.

For example, if we consider 20 and 42, their product is 20 × 42 = 840. However, both have a common factor of 2. Dividing 840 by 2 yields 420, which is the LCM of 20 and 42.

In fact, the equation below clarifies it more:

Product of two numbers = Product of their HCF (or GCD) and LCM

This can be rewritten as:

LCM = Product/HCF

Therefore, if the HCF of the two numbers is 1, their LCM is their product; otherwise, it is not.