If t is even and xy and z have no common factor show that t2

If t is even and x,y, and z have no common factor, show that t^2 =x^2+y^2+z^2 is impossible

Solution

if x,y,z have no common factor, then they must definitely be prime number...Eg-2,3,5,7,11,13,17,19,23,etc...

so, except 2 all primes are odd number

square of odd number=odd

square of even number =even

if,x or y or z ,any one is 2-even

then,x2+y2+z2 = even but not perfect square hence not t2

if,x or y or z ,all odd

then,x2+y2+z2 = odd hence not t2-even