Find all pairs of positive integers whose greatest common divisor is 12 and whose least common multiple is 144, with order not counted — that is, (12, 144) and (144, 12) count as one pair.

The pairs (12, 144) and (36, 48).

For any two positive integers, \gcd(a,b) \cdot \operatorname{lcm}(a,b) = a \cdot b.

Two Numbers With a Small GCD and a Large LCM

Issue 31 Core Published 23 Oct 2026 By The Chalkboard Desk

The problem

Find all pairs of positive integers whose greatest common divisor is and whose least common multiple is , with order not counted — that is, and count as one pair.

Both numbers are positive integers.
Only unordered pairs are requested.

Take a minute

Reveal hint 1

For any two positive integers, .

Reveal hint 2

Write , with and see what the lcm condition becomes.

Solution

AnswerThe pairs and .

Derivation

Use the product identity.
Change variables.
List coprime factor pairs.
Convert back.
Verify with prime exponents.
Confirm the count.