Prime Factorization and LCM

Question

Using prime factorization, how can you determine whether the least common multiple of two numbers is the product of the two numbers, or is less than the product of the two numbers?

Answer 1

take a specific example

12 and 16 both share a common prime factor of 2. LCM = 48 which is less than 12 x 16 = 192

if they have no common prime factor, then LCM = their product

if they share a prime factor LCM is less than their product

12 = 3 x 2 x 2

16 = 2 x 2 x 2 x 2

12 = 3 x 2^2, 16 = 2^4

take the primes, 3 and 2 and give them the highest power, which is an implicit 1 for 3 and 4 for 2, 3^1 and 2^4

3^1 = 3, 2^4 = 16

then multiply them together: 3 x 16 = 48