[Solved] How do I find the least b from lcm(a,b) = c?

Question

You can find the least common multiple using prime factorization. Here, lcm(a, b) has to contain all the prime factors of a and b in their highest multiple they appear in in any of the two numbers.

For example, 8 = 2^3 and 12 = 2^2 * 3, so lcm(8, 12) = 2^3 * 3 = 24.

This can easily be reversed: Find the prime factors of c (including their multiplicity), then check which ones already are covered by a. b has to be the product of the remaining ones.

So if c = 24 = 2^3 * 3 and a = 6 = 2 * 3, then b has to be 8 = 2^3. The 3^1 is already covered by a, but a only has 2^1, so b has to be 2^3.

Accepted Answer

You can find the least common multiple using prime factorization. Here, lcm(a, b) has to contain all the prime factors of a and b in their highest multiple they appear in in any of the two numbers.

For example, 8 = 2^3 and 12 = 2^2 * 3, so lcm(8, 12) = 2^3 * 3 = 24.

This can easily be reversed: Find the prime factors of c (including their multiplicity), then check which ones already are covered by a. b has to be the product of the remaining ones.

So if c = 24 = 2^3 * 3 and a = 6 = 2 * 3, then b has to be 8 = 2^3. The 3^1 is already covered by a, but a only has 2^1, so b has to be 2^3.