# Is it possible that in a Pythagorean triple only 2 values in (a, b, c) share a common factor greater than 1.

In a Primitive Pythagorean Triple (PPT) the set $$(a, b, c)$$ has GCD of 1.

I have read in the wiki page that every other Pythagorean triple can be obtained by multiplying k to this set $$(ka, kb, kc)$$ where $$k$$ is some positive integer.

I was wondering if the above statement is true then it's not possible for only 2 values in any Pythagorean triple to have a common factor greater than 1. Since the common factor will be forced to divide even the third value.

If I am not wrong in asking the question taking this equation

$$a^2 + b^2 = c^2$$

how to prove the above fact?

If $$d=(a,b),\dfrac aA=\dfrac bB=d$$(say)
$$c^2=d^2(A^2+B^2),\dfrac{c^2}{d^2}=A^2+B^2$$ which is an integer
$$\implies d^2\mid c^2\implies d\mid c$$
Similarly if $$e=(a,c)$$