Question

Use division algorithm to show that the square of any positive integer is of the form 3p,3p+1

Answer 1

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

To find :  Prove

Solution:

Any number can be represented  as

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

(3q)²

= 9q²  

= 3* 3q²  

= 3p    ( as q is integer => 3q²  is integer)

(3q + 1)²

= 9q² + 6q  + 1

= 3( 3q²  + 2q)  + 1

3q²  + 2q  is an integer as q is integer

= 3p + 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

= 3p + 1

Hence proved square of any integer is of the form 3p or 3p+1

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