Let ‘a’ be any positive integer.

Then,

According to Euclid’s division lemma,

a=bq+r

According to the question, when b = 4.

\[a=4k+r,n<r<4\]

When r=0, we get a=4k

\[\to {{a}^{2}}=16{{k}^{2}}=4\left( 4{{k}^{2}} \right)=4q,whereq=4{{k}^{2}}\]

When r=1, we get a=4k+1

\[\to {{a}^{2}}={{\left( 4k+1 \right)}^{2}}=16{{k}^{2}}+1+8k=4\left( 4k+2 \right)+1=4q+1\]

Where q\[=k\left( 4k+2 \right)\]

When r=2, we get a=4k+2

\[\to {{a}^{2}}={{\left( 4k+2 \right)}^{2}}=16{{k}^{2}}+4+16k=4\left( 4{{k}^{2}}+4k+1 \right)=4q\]

\[q=4{{k}^{2}}+4k+1\]

\[\to {{a}^{2}}={{\left( 4k+3 \right)}^{2}}=16{{k}^{2}}+9+24k=4\left( 4{{k}^{2}}+6k+2 \right)+1\]\[=4q+1,\]

\[q=4{{k}^{2}}+6k+2\]

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