2

I'm not sure if this question should be under Modular Arithmetic, but that's where it was in my book.

Show that, if $x$, $y$, and $z$ and integers such that $x^2 + y^2 = z^2$, then at least one of $\{x, y, z\}$ is divisible by $2$, at least one of $\{x, y, z\}$ is divisible by $3$ and at least one of $\{x, y, z\}$ is divisible by $5$.

What I did so far is to write them as:

$$x = 2k , \quad y = 3m , \quad z = 5n$$

Then the equation becomes $4k^2 + 9m^2 = 25n^2$ This would work for $k=2$, $m=1$, $n=1$, which would give $x=4$, $y=3$, $z=5$

I don't know how to show that it would only work for these numbers, or multiples of them.

Michael Hardy
  • 1
Anonymous
  • 77

1 Answers1

1

If $x^2$ and $y^2$ are both odd $z^2$ is even.

If $x^2$ and $y^2$ are not multiples of $3$ they are $1 \bmod 3$ so $z^2$ is $2\bmod 3$, which is not a quadratic residue.

If $x^2$ and $y^2$ are not multiple of $3$ they are either $1$ or $-1 \bmod 5$. So $z^2$ is $0,2$ or $3\bmod 5$. $2$ and $3$ are impossible because they are not quadratic residues.

Asinomás
  • 105,651