This is a problem from "Introduction to Mathematics - Algebra and Number Systems" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.
The problem reads as follows:
If 3p2 = q2, where p,q∈Z, show that 3 is a common divisor of p and q.
I am able to show that 3 divides q, simply by rearranging for p2 and showing that
p2∈Z⇒q2/3∈Z⇒3|q
However, I'm not sure how to show that 3 divides p.
Edit:
Moron left a comment below in which I was prompted to apply the solution to this question as a proof of √3's irrationality. Here's what I came up with...
[incorrect solution...]
...is this correct?
Edit:
The correct solution is provided in the comments below by Bill Dubuque.
Answer
Write q as 3r and see what happens.
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