A positive integer is of the form 3 q + 1 , q being a natural number. Can you write its square in any form other than 3 m + 1 i.e., 3 m or 3 m + 2 for some integer m? Justify your answer.

Foundation

Mathematics Foundation

Properties of Real Number

Linear Equations in One Variable

Question

A positive integer is of the form

3 q + 1, q

being a natural number. Can you write its square in any form other than

3 m + 1

i.e.,

3 m

3 m + 2

for some integer

m

? Justify your answer.

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

By Euclid's division algorithm ,

a = b q + r

where

a, b, q, r

are non-negative integers and

0 \leq r &lt; b

On putting

b = 3

and

r = 1

we get

a = 3 q + 1

Squaring both sides

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

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

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

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

, where

m = 4q^2 + 2q $ is any integer.</div></div><div><br/></div><div><br/></div><div>Hence, the square of a positive integer iof the form

3q+1

c a n n o t b e w r i t t e n i n a n y f o r m o t h e r t h a n

3m+1$$