Question

Let Euclids division lemma to show that the square of any positive integer is either of the form $3m$ or $3m+1$ for some integer $m$.[Hin: Let $x$ be any positive integer then it is of the form $3q,3q+1$ or $3q+2$. Now square each of these and show that they can be rewritten in the form $3m$ or $3m+1$

Answer 1

Let $a$ be any positive integer and $4b=3$

then by Euclid's division lemma

$a=3q+r$

$a=3qor3q+1$ or $3q+2$

$a^2=9q^2=3\left(3q^2\right)=3m,wherem=3q^2$

(1) If $a=3q$

$a^2=9q^2=3\left(3q^2\right)=3m,wherem=3q^2$

(2) if $a=3q+1$

$a^2={\left(3q+1\right)}^2=9q^2+1+6q$

$=9q^2+6q+1$

$=3\left(3q^2+2q\right)+1$

$=3m+1wherem=3q^2+2q$

(3) If $a=3q+2$

$a^2={\left(3q+2\right)}^2=9q^2+4+12q$

$=9q^2+12q+3+1$

$=3\left(3q^2+4q+1\right)+1$

$=3m+1,wherem=3q^2+4q+1$

Hence square of any positive integer is either of the form $3m$ or $3m+1$ for some integer $m$.