How do you determine if something is a multiple of a number using prime factors?

Question

I am reviewing some GRE/GMAT math and I don't have the books on me... but I'm trying to remember how prime factors related to multiples of a number.

Say we are trying to figure out if a number is a multiple of 10.

The prime factorization of 10 is $5^1 2^1$. So any multiple of 10 has to have at least a single 2 and a single 5 in its prime factorization right?

Let's take the number 20. That has a prime factorization of $2^2 5^1$ so it is a multiple of 10.

Let's take the number 38. That has a prime factorization of $19^1 2^1$. It is not a multiple of 10 because it does not share all the prime factors of 10 right? It is missing a single 5 right?

So if we then just add the 5 in its prime factorization $19^1 5^1 2^1$ which is the number 190, that is now a multiple of 10.

Is this how it works? What's the reason?

Answer 1

Yes. Alternatively, a number $x$ is a multiple of $10$ if we may write $x=10k$ for an integer $k$.