Use Euclid’s division lemma to show that the square of any positive integer is e...

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September 2021 1 14 Report

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. Please explain in detail.

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Sudin

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As per Euclid's Division Lemma
If a & b are 2 positive integers, then
$a = bq + r$
$where \: 0 \leqslant r < b$
Let positive integer be a
And b = 3
$hence \: a = 3q + r$
$where(0 \leqslant r < 3)$
r is an integer greater than or equal to 0 and less than 3
hence, r can be either 0, 1 or 2.

CASE 1:-
$if \: r = 0 \\ our \: equation \: becomes \\ a = 3q + r \\ a = 3q + 0 \\ a = 3q \\ squaring \: both \: the \: sides \\ {a}^{2} = {(3q)}^{2} \\ {a}^{2} = 9 {q}^{2} \\ {a}^{2} = 3(3 {q}^{2} ) \\ {a}^{2} = 3m \\ where \: m = 3 {q}^{2}$

CASE 2:-
$if \: r = 1 \\ our \: equation \: becomes \\ a = 3q + r \\ a = 3q + 1 \\ squaring \: both \: the \: sides \\ {a}^{2} = {(3q + 1)}^{2} \\ {a}^{2} = {(3q)}^{2} + {1}^{2} + 2(3q) \\ {a}^{2} = 9 {q}^{2} + 6q + 1 \\ {a}^{2} = 3 ({3q}^{2} + 2q) + 1 \\ {a }^{2} = 3m + 1 \\ where \: m \: = 3 {q}^{2} + 2q$

CASE 3:-
$if \: r \: = 2 \\ our \: equation \: becomes \\ a = 3q + r \\ a = 3q + 2 \\ squaring \: both \: the \: sides \\ {a}^{2} = {(3q + 2)}^{2} \\ {a}^{2} = {(3q)}^{2} + {2}^{2} + 2(2)(3q) \\ {a}^{2} = 9 {q}^{2} + 12q + 4 \\ {a}^{2} = 9 {q}^{2} + 12q + 3 + 1 \\ {a}^{2} = 3(3 {q}^{2} + 4q + 1) + 1 \\ {a}^{2} = 3m + 1 \\ where \: m = 3 {q}^{2} + 4q + 1$
Hence, square of any positive number can be expressed of the form 3m or 3m + 1

HENCE PROVED
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