Chinese Remainder Theorem Problem

**hilly** · December 21st 2009, 10:40 AM

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.

**Bruno J.** · December 21st 2009, 01:49 PM

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 $f(a) \equiv 0 \mod n$ .

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