1650.31 – Divisibility Theorem (Sum)

Suppose $x$ , $y$ , and $z$ are all counting numbers, and $x + y = z$ . If two of these numbers are multiples of 7, is the third one also a multiple of 7? Show why or why not.

Solution

The answer is yes; here's why:

A multiple of 7 can be written $7k$ , or $7n$ , or whatever, where $k$ , $n$ , etc. are whole numbers. So if $x + y = z$ , and $x$ and $y$ are multiples of 7, we can say $x$ = $7k$ and $y$ = $7n$ .

Then $z = x + y = 7k + 7n = 7(k + n)$ , and $z$ is also a multiple of 7. The argument is the same for the other cases, since $x = z - y$ and $y = z - x$ .