Is the Least Common Multiple to two prime numbers always their product? Why or Why not?


Hi Kristine. A common multiple of two numbers is a composite number that both numbers can divide evenly. That means that if those numbers have no prime factors in common, then indeed, their product is the smallest common multiple of the two.

Compare this to two non-prime numbers: 4 and 6. The prime factors of 4 are 2 and 2, the prime factors of 6 are 2 and 3. So the least common multiple will be the maximum number of factors of 2 (that's two) and the maximum number of factors of 3 (that's one). So it is 2 x 2 x 3 = 12.

For a harder example, consider 84 and 315. What is their LCM? It's not as hard as it looks. The prime factors are 84 = 2 x 2 x 3 x 7 and 315 = 3 x 3 x 5 x 7. So select out the maximum unique factors: two 2's, two 3's, one 5 and one 7. 2 x 2 x 3 x 3 x 5 x 7 = 1260. To check, 1260/84 = 15 and 1280/315 = 4.

A similar thing happens when one of the numbers is prime and the other is composite. You can test this out for yourself with 11 and 55. The product is 605, but the LCM is less than this. What is it?

Hope this helps,
Stephen La Rocque>