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  • Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Use Euclid’s division lemma to show that the square of any positive integer is either of the form $3m$ or $3m+1$ for some integer $m$.
[Hint : 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 $3m+1$.]

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By Euclid division lemma,we know that
If a and b are two positive integers, then,by euclid division lemma 
$a = bq + r, 0 \leq r \leq b$
Let $b = 3$
Therefore, $r = 0, 1, 2$
Therefore, $a = 3q$ or $a = 3q + 1$ or $a = 3q + 2$
If $a = 3q$: 
$a^{2}=9q^{2}=3(3q^{2})$
If $a = 3q + 1$ :
$a^2=9 q^2+6 q+1=3\left(3 q^2+2 q\right)+1=3 m^2+1$ 
If $a = 3q + 2$ :
$a^2=9 q^2+12 q+4=3\left(3 q^2+4 q+1\right)+1=3 m^2+1$
Therefore, the square of any positive integer is either of the form $3m$ or $3m + 1$. 
Posted by

Pankaj Sanodiya

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