For all integers m, prove the following proposition below: m2=5k or m2=5k+1 or m2=5k+4 Solution Suppose m=5n, where n is an integer. Then we have m^2=25n^2 and k=5n^2 and we have the first case. Suppose m=5n+1, we have m^2=25n^2+10n+1=5(5n^2+2n)+1 and we are in case two. Suppose m=5n+2, we have m^2=25n^2+20n+4=5(5n^2+4n)+4 which is case three. Suppose m=5n+3, we have m^2=25n^2+30n+9=5(5n^2+6n+1)+4 which is case three Suppose m=5n+4, m^2=25n^2+40n+16=5(5n^2+8n+3)+1 which is case two. And this yields all possibilities..