Prove that the square of any positive integer is of the form `4q` or `4q+1` for some integer `q` .

Explanation

Let a be an arbitary positive integer. Then, by Euclid division alogorithm, corresponding to the positive intergers 'a' and 4, there exist non negative integer m and r such that
a=4m+r, where `0 le r lt 4`
`a^(2)=16m^(2)+r^(2)+8mr` ….(i)
where, `0 le r lt 4` `[therefore(a+b)^(2)=a^(2)+2ab+b^(2)]`
CASE I When r=0, then putting r=0 in Eq. (i), we get
`a^(2)=16m^(2)=4(4m^(2))=4q`
where, q=`4m^(2)` is an integer.
CASE II When r=1, then puttting r=1 in Eq. (i), we get
`a^(@)=16m^(2)+1+8m`
`=4(4m^(2)+2m)+1=4q+1`
Where, q=`(4m^(2)+2m)` is an integer.
CASE III When r=2, then putting r=2 in Eq. (i) we get
`a^(2)=16m^(2)+4+16m`
`=4(4m^(2)+4m+1)=4q`
where, q=`(4m^(2)+4m+1)` is an integer
CASE IV When r=3, then putting r=3 in Eq. (i), we get
`a^(2)=16m^(2)+9+24=16m^(2)+24m+8+1`
`=4(4m^(2)+6m+2)+1=4q+1`
where, q=`(4m^(2)+6m+2)` is an integer
Hence the square of the positive integer is either of the form 4q or 4q+1 for some integer q.