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

Question:

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

Solution:

By Euclid's lemma, $b=a q+r, 0 \leq r Here, b is a positive integer and = 3.$\therefore b=3 q+r$, for$0 \leq r<3$This must be in the form$3 q, 3 q+1$or$3 q+2$. Now,$(3 q)^{2}=9 q^{2}=3 m$, where$m=3 q^{2}(3 q+1)^{2}=9 q^{2}+6 q+1=3\left(3 q^{2}+2 q\right)+1=3 m+1$, where$m=3 q^{2}+2 q(3 q+2)^{2}=9 q^{2}+12 q+4=3\left(3 q^{2}+4 q+1\right)+1=3 m+1$, where$m=3 q^{2}+4 q+1\$

Therefore, the square of a positive integer 3q+1 is always in the form of 3m or 3m+1 for some integer m.