Write whether the square of any positive integer can be of the form 3 m + 2, where m is a natural number. Justify your answer.

Foundation

Mathematics Foundation

Properties of Real Number

Linear Equations in One Variable

Playing with Numbers

Question

Write whether the square of any positive integer can be of the form

3 m + 2

, where

m

is a natural number. Justify your answer.

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By Euclid's division lemma,

b = a q + r

where

a, b, q, r

are

+ v e

integers and here

a = 3

then

b = 3 q + r

then

0 \le r &lt; 3

r = 0, 1, 2

b

becomes

b = 3 q, 3 q + 1, 3 q + 2

b = 3 q

\Rightarrow (b)^{2} = (3 q)^{2} 2

\Rightarrow, b^{2} = 3.3 q^{2} = 3 m

where,

3 q^{2} = m

So, as

b^{2}

is perfect square so

3 m

will also be perfect square.

when

r = 1, b = 3 q + 1

\Rightarrow (b)^{2} = (3 q + 1)^{2}

\Rightarrow b^{2} = 9 q^{2} + 1 + 2 \times 3 q

\Rightarrow b^{2} = 3 [3 q^{2} + q] + 1

\Rightarrow b^{2} = 3 m + 1

and

m = 3 q^{2} + 2 q

So,

b^{2}

is perfect square or a number of the form

3 m + 1

is perfect square.

when

r = 2, b = 3 q + 2

\Rightarrow b^{2} = 9 q^{2} + 4 + 2.3 q . 2

= 9 q^{2} + 3 + 3 \times 4 q + 1

= 3 [3 q^{2} + 1 + 4 q] + 1

\Rightarrow b^{2} = 3 m + 1

Again, a number of the form

3 m + 1

is perfect square.

Hence, a number of the

(3 m + 2) c m

never be perfect square. But a number of the form

3 m

, and

3 m + 1

are perfect squares.