A positive integer is the form of 3q+1 q, being a natural number. Can you write its square in any form other than 3m+1 i.e. 3m or 3m+2 for some integer? Justify your answer.

Explanation

No, by Euclid Lemma, b=aq+rm `0 le r lt a`
Here, b is any positive integer `a=3, b=3q+r for `0 le r lt 3`
So, this must be in the form `3q,3q+1 or 3q+2`
Now, `(3q)^(2) =9q^(2)=3m` [here, `m=3q^(2)`]
and `(3q+1)^(2)=9q^(2)+6q+1`
`=3(3q^(2)+2q)+1=3m+1` [where, `m=3q^(2)+2q`]
Also, `(3q+2)^(2)=9q^(2)+12q+4`
`=9q^(2)+12q+3+1`
`3=(3q^(2)+4q+1)`
`3m=1` [here, `m=3q^(2)+4q+1]`
Hence, square of a positive integer is of the form 3q+1 is always in the form 3m+1 for some integer m.