Question

Two non-negative integers are chosen at random. The probability that the sum of the square is divisible by 5 is

• A. $9/16$
• B. $9/25$
• C. $9/17$
• D. None of these

Answer 1

The correct option is B $9/25$

Let $x=5m+r,y=5n+5$
where $m,n\in I,0\le r,s\le 4$
$\Rightarrow x^2+y^2=25(m^2+n^2)+10(mr+ns)+r^2+s^2$
The ordered pair $(r,s)$ can take $5\times 5=25$ values.
But $r^2+s^2$ is divisible by $5$ if $(r,s)=(0,0),(1,2),(2,1),(2,2),(1,3),(3,1),(2,4),(4,2),(3,4),(4,3)$