If $(a,b,c)$ is a primitive Pythagorean triplet, explain why... If $(a,b,c)$ is a primitive Pythagorean triplet, explain why only one of $a$,$b$ and $c$ can be even-and that $c$ cannot be the one that is even.
What I Know:
A Primitive Pythagorean Triple is a Pythagorean triple $a$,$b$,$c$ with the constraint that $\gcd(a,b)=1$, which implies $\gcd(a,c)=1$ and $\gcd(b,c)=1$. Example: $a=3$,$b=4$,$c=5$  where, $9+16=25$
At least one leg of a primitive Pythagorean triple is odd since if $a$,$b$ are both even then $\gcd(a,b)>1$

Clearly they cannot all be even as a smaller similar triple could be obtained by dividing all the sides by $2$ (your final point).
Nor can two of them be even since $a^2+b^2=c^2$ and either you would have an even number plus an odd number (or the other way round) adding to make an even number or you would have an  even number plus an even number adding to make an odd number.
Nor can they all be odd since $a^2+b^2=c^2$ and you would have an odd number plus an odd number adding to make an odd number.
Added:  Nor can the two shorter sides be both odd, say $2n+1$ and $2m+1$ for some integers $n$ and $m$, as the longer side would be even, say $2k$ for some integer $k$,  as its square is the sum of two odd numbers,  but you would then have $4n^2+4n+1 +  4m^2+4m+1 = 4k^2$ which would lead to $k^2 = n^2+n+m^2+m + \frac{1}{2}$ preventing $k$ from being an integer.
