Answer
See below.
Work Step by Step
Based on the given conditions, we have m mod 5= 0, 1, 2, 3, 4
Case1: m mod 5= 0, we can write m=5n, thus $m^2=25n^2=5(5n^2)$;
Case2: m mod 5= 1, we can write m=5n+1, thus $m^2=25n^2+10n+1=5(5n^2+2n)+1$;
Case3: m mod 5= 2, we can write m=5n+2, thus $m^2=25n^2+20n+4=5(5n^2+4n)+4$;
Case4: m mod 5= 3, we can write m=5n+3, thus $m^2=25n^2+30n+9=5(5n^2+6n+1)+4$;
Case5: m mod 5= 4, we can write m=5n+4, thus $m^2=25n^2+40n+16=5(5n^2+8n+3)+1$;
thus we have covered all the cases for any integer $m$ for the given statement to be true.
