Question

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

NCERT Class X
Mathematics - Exemplar Problems

Chapter _Real Numbers

Answer 1

⇒ No ,  square of any positive integer of the form 3m + 1 is always in the form in the 3m + 1 , but it'snot in the form of - Either 3m or 3m + 2  because of the following solution :

⇒ a = bq + r 

Let ''a'' be any positive integer and 'q be the quotient and let ''r'' be the remainder .

Therefore we get ,

a = ( 3 q + 1 )² { We square it  , as according to the question }

a = 9 q² + 6 q² + 1² { (a + b )² = a² + b² + 2 a b }

a = 3 ( 3 q² + 2 q²) + 1

a = 3 q + 1 ( Where , q is 3 q + 1 )

So, we  got that it is always in the form of 3 q + 1.

Answer 2

Solution:No,the square of any positive integer in form of 3m+1 will have 3q+1 as it's square let's look at an example:

We know ,0=<r<b

Therefore in case of 3m+1 (0<=r<3)

(3m+1)2=9m2+1+6m

=9m2+6m+1

=3(3m2+2m)+1{where 3m2+2m=q}

=3q+1

We can say that square of positive integer in form of 3m+1 is 3q+1