Q

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  r  b Let b = 3

Therefore, r = 0, 1, 2

Therefore, a = 3q or a = 3q + 1 or a = 3q + 2

If a = 3q:

$a^2=9q^2=3(3q^2)=3m^2$

If a = 3q + 1 :

$a^2=9q^2+6q+1=3(3q^2+2q)+1=3m^2+1$

If a = 3q + 2 :

$a^2=9q^2+12q+4=3(3q^2+4q+1)+1=3m^2+1$

Therefore, the square of any positive integer is either of the form 3m or 3m + 1.

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