Question

Prove that square of any positive integer is of the form 5q,5q+1,5q+4 for some integer q

Answer 1

By Euclid’s division algorithm 
a = bm + r, where 0 ≤ r ≤ b 
Put b = 5 
a = 5m + r, where 0 ≤ r ≤ 4 
If r = 0, then a = 5m 
If r = 1, then a = 5m + 1 
If r = 2, then a = 5m + 2 
If r = 3, then a = 5m + 3 
If r = 4, then a = 5m + 4 
Now, (5m)² = 25m² 
= 5(5m²) 
= 5q where q is some integer

Hence, the square of any positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q.