@neauoire Are you familiar with how positive integers can be written in the form M = p1^e1 * p2^e2 * ... and N = p1^f1 * p2^f2 * ... where p1, p2, ... are the primes?

A related equation is min(M, N) + max(M, N) = M + N.

If we look at a single prime pi, its exponent is
min(ei, fi) in gcd(M, N),
max(ei, fi) in lcm(M, N)
ei +fi in M*N.

For example with M = 4 = 2^2 * 3^0 and N = 6 = 2^1 * 3^1:

gcd(M, N) = 2 = 2^1 * 3^0
lcm(M, N) = 12 = 2^2 * 3^1
M * N = 24 = 2^(2 + 1) * 3^(0 + 1)