1650.31 – Divisibility Theorem (Sum)

Tags:Problem Set 3GeometrySymbol-Pushing

Suppose xx, yy, and zz are all counting numbers, and x+y=zx + 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.


The answer is yes; here's why:

A multiple of 7 can be written 7kk, or 7nn, or whatever, where kk, nn, etc. are whole numbers.

So if x+y=zx + y = z, and xx and yy are multiples of 7, we can say xx = 7kk and yy = 77n.

Then z=x+y=7k+7n=7(k+n)z = x + y = 7k + 7n = 7(k + n), and zz is also a multiple of 7.

The argument is the same for the other cases, since x=zyx = z - y and y=zxy = z - x .