$a \equiv 3 \pmod{5}$ and $b \equiv 4 \pmod{5}$. What is $(a^{2} + 2b) \bmod 5$?

Explanation

1. 01

   Replace $a$ and $b$ by their remainders: $a^{2} + 2b \equiv 3^{2} + 2 \times 4 = 17 \pmod{5}$.

2. 02

   $17 = 5 \times 3 + 2$, so the answer is 2.