When is the least common multiple of two positive integers equal to their product?

Conditions

• The two integers are positive.
• The two integers are coprime (gcd(a,b) = 1).

Reasoning, step by step

1. Check if the two integers share any common divisors greater than 1.
2. If they do not share any common divisors other than 1, they are coprime.
3. For coprime integers, the LCM is simply the product of the two numbers.

Example

The video states, 'LCM(a, b) = $a*b$ if a and b are coprime,' and gives the example LCM(5, 12) = 60. It contrasts this with 4 and 6, which are not coprime, noting their product is 24 but their LCM is only 12.

Common misconceptions

• Assuming LCM(a,b) = $a*b$ for all pairs of positive integers.
• Forgetting that non-coprime moduli require using the true LCM or the GCD formula, otherwise valid solutions might be skipped.

Watch the explanation

Explore next

• What is the general formula for LCM(a,b) using GCD?
• Why does using the product instead of the LCM for non-coprime moduli cause errors in CRT?
• How do you compute LCM(4,6) by listing multiples?