Thread: division question

Let x, y, d, be elements of Z (integers) and let z = x+y. Prove: If d divides any two of x, y, z then d divides the third.

How do you prove this? I know it's true, but not how to prove it.

Originally Posted by vassago

Let x, y, d, be elements of Z (integers) and let z = x+y. Prove: If d divides any two of x, y, z then d divides the third.

How do you prove this? I know it's true, but not how to prove it.

Suppose that $\displaystyle d|x \implies x=q_1d$ and $\displaystyle d|y \implies y=q_2d$

Now sub into the above equation and you get

$\displaystyle z=q_1d+q_2d \iff z=(q_1+q_2)d \implies d|z$