The sum of three multiples of 7 is a multiple of 7

Statement

Prove that the sum of three integers divisible by $7$ is divisible by $7$.

Solution

Let $a$, $b$, $c$ be three integers divisible by $7$. Then there are integers $x$, $y$, $z$ with $a = 7x, \quad b = 7y, \quad c = 7z.$ Adding: $a + b + c = 7x + 7y + 7z = 7\,(x + y + z).$ Since $x+y+z$ is an integer, $a+b+c$ is a multiple of $7$. $\blacksquare$