Question

Prove that if (x,y,z) is a Pythagorean triple then at least one of x, y

is divisible by 3.​

Answer 1

Answer:

We know that 5 divides at most one of x,y and z

If 5 does not divide x or y, then x2≡±1(mod5) and y2≡±1(mod5)

Then z2≡0,2 or −2(mod5)

But ±2 is not a quadratic residue modulo 5

So z2≡0(mod5), whence 5 ∣z.