Congruence is compatible with addition

Statement

Let $n$ be a positive integer, and let $a$, $b$, $c$ and $d$ be integers. If $a \equiv b \pmod n$ and $c \equiv d \pmod n$, then $a + c \equiv b + d \pmod n$.

Remarks

The proof: $(a + c) - (b + d) = (a - b) + (c - d)$, and $n$ divides both terms. The same holds for subtraction: $a - c \equiv b - d \pmod n$. Together with compatibility with multiplication, this result shows that the addition of integers modulo n is well defined.

Explanations

L1 human

To find the remainder of a sum, you can add the remainders and reduce again. For example, $17 \equiv 2 \pmod 5$ and $24 \equiv 4 \pmod 5$, so $17 + 24 = 41 \equiv 2 + 4 = 6 \equiv 1 \pmod 5$.