---
tag: 00FWBN7F0Y
kind: statement
role: proposition
name: Congruence is compatible with addition
msc: [11A07]
uses: [0TPW25CBN6]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_arithmetic
    locator: Basic properties
---

## 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|072D5G4SK6]], this result shows that the addition of [[integers modulo n|01N3AYAA6N]] 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 $17 + 24 = 41 \equiv 2 + 4 = 6 \equiv 1 \pmod 5$.