# Chinese Remainder Theorem Problem

• December 21st 2009, 10:40 AM
hilly
Chinese Remainder Theorem Problem
I'm not quite sure how to start on this problem.

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

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

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

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

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

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

Do I use the actual solution $x_i$ of $f(x)\equiv 0\mod n_i$, or just the fact that it has a solution. Sorry, I'm just lost on this problem.
• December 21st 2009, 01:49 PM
Bruno J.
For $i=1,2,3$ let $f(x_i) \equiv 0 \mod n_i$. Apply the CRT to find an integer $a \equiv x_i \mod n_i$ for $i=1,2,3$. Then $f(a) \equiv f(x_i) \equiv 0 \mod n_i$ for $i=1,2,3$ so $f(a) \equiv 0 \mod n$.