Question

Show that square of any odd positive int can be of the form 6q+1 or 6q+3

Answer 1

Answer:

Let 'a' be any positive integers on dividing 'a' by 6 we get quotient as q and remainder as r .

By EDL ,

a=6q +r

R=0,1,2,3,4,5.

put the value of r , we get

a=6q, 6q+1,6q+2,6q+3, 6q+4, 6q+5.

Case1- a= 6q

here 6 is divisible by 2 so it is an even positive integer

Case2-a=6q +1

here 6 is divisible by 2 ,but 1 is not divisible by 2 so it is an odd positive integer .

Case 3- a= 6q+ 2

here 6and 2 both are divisible by 2 so it is an even positive integer.

Case 3-a=6q+3

here 6 is divisible by 2 but 3 is not divisible bye 2 so it is an odd positive integer.

Case 4 a=6q+4

here 6 and 4 both are divisible by 2 so it is an even positive integer

Case 5 a=6q+5

here 6 is divisible by 2 but 5 is not divisible by 2 so it is an odd positive integer

we can see that the odd positive integer are 6q+1,6q+3 .