Question

Two numbers x and y are such that when divided by 6 they leave remainders 4 and 5 respectively Find the remainder when $(x^2+y^2)$ is divided by 6

Answer 1

Suppose $x=6k_1+4$ and $y=6k_2+5$
$x^2+y^2=(6k_1+4{)}^2+(6k_2+5{)}^2$
$=36k{{}_1}^2+48k_1+16+36k{{}_2}^2+60k_2+25$
$=36k{{}_1}^2+48k_1+36k{{}_2}^2+60k_2+41$
Obviously when this is divided by 6 the remainder will be 5