I've been having some issues with the following problem for a few days, and I think I might have found the answer, but I'm not quite sure:
Problem: Let f(x) be a polynomial with integer coefficients. Assume integer $m$ exists such that $f(m) \equiv 0\ (mod \ 10) $ and integer $n$ exists such that $f(n) \equiv 0\ (mod \ 33)$. Integers $m$ and $n$ do not have to be equal. Must there exist some integer $q$ such that $f(q) \equiv 0\ (mod\ 330)$? Why or why not?
I understand that, by principle, the equation f(x) would have to include 330 as a possible divisor considering it contains the components that generate both 10 and 33, but I'm confused as to why that is. Am I able to use the Chinese Remainder Theorem to say that since all of the congruences are using the same function of $f(x)$, by the CRT, there must exist an integer $q$ such that $f(q) \equiv 0\ (mod\ (33*10))$?
Any help would be much appreciated!