# Two little number theory questions

First question: Let $a$, $b$, $c$ be distinct primes larger than $3$, and let $x = abc$. Show that if $p$ is a positive integer and $p^2 \equiv 9 \pmod {x}$, then $p \equiv \pm 3 \pmod {a}$, $p \equiv \pm 3 \pmod{b}$, and $p \equiv \pm 3 \pmod{c}$.

Second question: Using the fact that $1001 = (7)(11)(13)$ find the eight solutions to $y^2 \equiv 9 \pmod{1001}$ in the set $\{ 1, 2, 3,\ldots, 1000 \}$.

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Homework? It looks like it... –  Guess who it is. Oct 10 '10 at 5:41
I have certainly set similar problems as homework in the past :-) –  Robin Chapman Oct 10 '10 at 8:10

HINT$\rm\ \ a|x|(p-3)(p+3)\ \Rightarrow \ a\:|\:p-3\ \: or\ \: a\:|\:p+3\:,\:$ since $\rm\:a\:$ is prime. Combine this with the Chinese remainder theorem to solve the second question. See also my answer here.