Show implication of x and y generates a remeinder

Question

I know:

1. If x is divided by 30 the remainder is 4.
2. If y is divided by 30 the remainder is 18.

Now the question is how do you show that x*y divided by 30 will give the remainder 12.

I know that I can brute force it, like so:

34/30 --> remainder will be 4, so x=34
48/30 --> remainder will be 18, so y=48

This gives:

(34*48)/30 --> remainder will become 12 which was to be proven.

However I'm wondering if there is a more "elegant" way of showing this, in a more logical way, rather than just guessing some numbers and putting them together?

Answer 1

$(30m+4)(30n+18)=900mn+540m+120n+72=30(30mn+18m+4n+2)+12$