<s>[INST] can the number $x^2 +y^2$ when with x and y positive integers 
can the number $x^2 +y^2$  when with $x$ and $y$ positive integers,
  end in $03$?

I know that $x^2+y^2$ can never end with unit digit 03 but am not sure how would I show the proof of that.
 [/INST] Let $x = 10a \pm b$ and $y = 10c \pm d$ where $b,d = 0,1,2,3,4$ or $5$.
So $x^2 + y^2 = 100(a^2 + c^2) + 20(\pm ab \pm cd) + (b^2 + d^2)$.
The only way for $b^2 +d^2$ to end in $3$ is if $b=2$ and $d=3$ (or vice versa)
So $x^2 + y^2 =100(a^2 + c^2) + 20(\pm 2a \pm 3c) + 13$
And the only we for that to end in $03$ is for $20(\pm 2a \pm 3c)+10$ to end in $00$ or for $2(\pm 2a \pm 3c)+1$ to end in $0$.   
But $2(\pm 2a\pm 3c) +1$ is odd so that can never happen.
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