Thread: expansion and factorisation diophantine equation

hey could someone please help me with this question... i think 10 is one of the answers buh i got that through guess and check and not through a proper mathematical method...

Find all the positive integer solutions to the equation

X^2 + y^2 + z^2 = 10(x+y+z)

Prove there are no other solutions


thankx,,,

We have,
x^2+y^2+z^2=10(x+y+z)
x^2+y^2+z^2=10x+10y+10z
x^2-10x+y^2-10y+z^2-10z=0
(x^2-10x+25)+(y^2-10y+25)+(z^2-10z+25)=75
(x-5)^2+(y-5)^2+(z-5)^2=75
a^2+b^2+c^2=75

By Fermat's 3 square theorem we can express it as a sum of three squares. We do that by trial an error (if you allow non-negatives),
0+25+36=75

Thus,
x=0,y=10,z=11
x=10,y=0,z=11
x=10,y=11,z=0
x=0,y=10,z=10
x=11,y=0,z=10
x=11,y=10,z=0