2025/029) Prove that if $x^2+xy+y^2$ is divisible by 10 it is divisible by 100

As 10 = 2 *5 let us check for both 2 and 5

We have $x^2+ xy+y^2 = x^2+ y(x+y)$

if x is odd the $x^2$ is odd and either y or x+y is even so the expression is odd 

similarly if y is odd.

So both x and y are even  and each term is divisible by 4 so the sum

let the expression be divisible by 5

working in mod 5 let $x=0$ we get $x^2+xy + y^2= y^2$ and it is divisible by 5 only if $y=5$ then each term is divisible by 25 so the expression.

Now let us have x is not zero .let $y = mx$

so we get $x^2+xy + y^2=  x^2(1+m+m^2)$

as x is non zero deviding by x we get $1+m+m^2$ ant for m = 0 to 4 we get it not a multiple of 5 so x and y has to be zero mod 5

So we have the expression divisible by 25 and also 4 so by 100

Proved