Find integer that satisfies two congruences Find the integer between $0$ and $29\times 23$ $= 667$ that satisfies the two following congruences: 
$x ≡ 15$ (mod $23$)
$x ≡ 1$ (mod $29$).

You can just look for a pattern. Each time you add $29$ to a number, the remainder when divided by $23$ increases by $6$, so you can see that: 
$$1≡1\pmod {23}, \;30≡7\pmod {23}, \;59≡13\pmod {23},\;\;...\;\;291≡15\pmod {23}$$ and there's your answer.
