Question

prove that square of any positive integer is of the form 9k or 9k+1.​

Answer 1

Given : square of any integer is of the form 9k or 3k+1.

To find :  Prove

Solution:

Any number can be represented  as

3q , 3q + 1 , 3q + 2   where k is integer

Lets find square of each case

(3q)²

= 9q²  

= 9k    ( as q is integer => q²  is integer)

(3q + 1)²

= 9q² + 6q  + 1

= 3q(3q + 2)  + 1

q(3q + 2)  is an integer as q is integer

= 3k + 1

(3q + 2)²

=  9q² + 12q  + 4

= 9q² + 12q  + 3 + 1

= 3( 3q²  + 4q + 1 )  + 1

3q²  + 4q + 1  is an integer as q is integer

= 3k + 1

Hence square of any integer is of the form 9k or 3k+1

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