Say I have a dividend (15) and a divisor (6). Using prime numbers, how do I tell if the dividend is divisible by the divisor?
The primes for 15 are $5^13^1$. The primes for 6 are $3^12^1$.
What's the rule regarding prime factors and divisibility? The divisor needs to have only the prime factors of the dividend, in any amount, but no other factors?
But a number is a multiple of another number if it contains at least the factorization of the smaller number? Is that correct? For example:
45 has a factorization of $3^25^1$
15 has a factorization of $3^15^1$
9 has a factorization of $3^2$
The above 3 are factors of 45.
But: 12 has a factorization of $2^23^1$
12 is not a factor because of the extra 2 right? 45 is not a multiple of 12 because it does not contain the factorization of 12 right?