Subjective Type

A positive integer is of the form $$3q+1, q$$ 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 $$m$$? Justify your answer.

Solution

By Euclid's division algorithm , $$ a = bq + r $$ where $$ a , b , q , r $$ are non-negative integers and $$ 0 \leq r < b$$.
On putting $$ b = 3$$ and $$r=1$$ we get
$$a = 3q + 1 $$
Squaring both sides
$$\Rightarrow a^2= (3q + 1)^2 $$
$$\Rightarrow a^2 = (3q)^2 + (1)^2 + 2(3q) $$
$$\Rightarrow a^2 = 3(3q^2 + 2q ) + 1 $$
$$ \Rightarrow a^2= 3m + 1 $$ , where $$ m = 4q^2 + 2q $$ is any integer.


Hence, the square of a positive integer iof the form $$3q+1$$ can not be written in any form other than $$3m+1$$

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