2.5.2, due on September 21

I thought I understood the Chinese remainder theorem, but when they used it to find the solution to example 2.5.2.1, I wasn't sure where they got the numbers that they were multiplying together. The theorem that says if f(x) has degree higher than p, there exists a polynomial of degree less than p where the solutions to the two functions are the same is a little bit hard for me to believe. I just can't think of why this would have to be true, but if i become convinced that it is, then it's a pretty cool concept.