# Thread: Chinese Remainder Theorem Problem

1. ## Chinese Remainder Theorem Problem

I'm not quite sure how to start on this problem.

Let $\displaystyle n_1,n_2,n_3$ be mutually prime where $\displaystyle n=n_1n_2n_3$.

Let $\displaystyle b_1,b_2,b_3$ be any integers,then there exists x such that $\displaystyle x\equiv b_i\mod n_i$.

Suppose $\displaystyle f(x)=7x^3-x^2+2x-11$.

Suppose $\displaystyle f(x)\equiv 0$ has a solution mod $\displaystyle n_i$ for $\displaystyle i=1,2,3$.

Prove that $\displaystyle f(x)\equiv 0$ has a solution mod n.

I don't need to find the solution, just show that one exists.

Do I use the actual solution $\displaystyle x_i$ of $\displaystyle f(x)\equiv 0\mod n_i$, or just the fact that it has a solution. Sorry, I'm just lost on this problem.

2. For $\displaystyle i=1,2,3$ let $\displaystyle f(x_i) \equiv 0 \mod n_i$. Apply the CRT to find an integer $\displaystyle a \equiv x_i \mod n_i$ for $\displaystyle i=1,2,3$. Then $\displaystyle f(a) \equiv f(x_i) \equiv 0 \mod n_i$ for $\displaystyle i=1,2,3$ so $\displaystyle f(a) \equiv 0 \mod n$.