Find a prime $p$ satisfying $p \equiv 1338 \mod 1115$

Question

Find a prime $p$ satisfying $p \equiv 1338 \mod 1115$. Are there infinitely many such primes. A little confused about this problem, any help or advice?

Answer 1

Hint: If $p \equiv 1338 \pmod{1115}$ then $p = 1338+1115n$ for some integer $n$.

Now, note that $1338 = 6 \cdot 223$ and $1115 = 5 \cdot 223$. Hence, $p = 223(6+5n)$.

What does this tell you about $n$?

Answer 2

Actually this is easy. Notice that $p \equiv 1338 (mod 1115)$ iif $p-0 \equiv 1338-1115 (mod 1115)$ because $1115 \equiv 0 (mod 1115)$. Also $1338-1115 = 223$ is prime.

Therefore 223 is the prime you're looking for.