TITLE: Product of three primes that is a square modulo 389
QUESTION [5 upvotes]: Find $n$ such that

$n$ is a product of three prime numbers and
$n$ is a square modulo $389$.


I'm not sure how to begin with this problem. Do I have to use an algorithm involving quadratic reciprocity? How can I apply it?

REPLY [6 votes]: By quadratic reciprocity, 2 and 3 are not squares modulo 389, but 5 is, so you can take $n=30$.