Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

By taking,’ a’ as  any positive integer and b = 3.
Applying Euclid’s  algorithm
a = 3q + r                
Here,
So, a = 3q or 3q+1 or 3q+2
And,
a2 = (3q)2 or (3q+1)2 or (3q+2)2
a2 = (9q2)  or  9q2+6q+1 or  9q2+12q+4
a2 = 3(3q2)2 or (3q2 + 2q)+1 or 3(3q2+4q+1)+1
a2 = 3k1 or 3k2+1 or 3k3+1
Where k1, k2 and k3 are some positive integers
Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.