What integer $n$ satisfies $0\le n<19$ and $$38574\equiv n\pmod{19}~?$$
I know that $38574\equiv n\pmod{19}$ means that $38574$ and $n$ have the same remainder when divided by $19$. So, I can try to find the remainder of $38574$ when divided by $19$ by using long division or a calculator. I get that $38574=2030\times 19+4$, so the remainder is $4$. That means $n$ must also be $4$, since it has the same remainder. I can check that $4$ satisfies the given inequality: $0\le 4<19$. Therefore, the integer $n$ that satisfies the given equation and inequality is $n=4$. 