Question

prove that the square of any positive odd integer is of the form 8m+1

Answer 1

Let a be any positive integer

When a is divided by 4, then possible remainders are o,1,2,3.

So when r=0

Then

a = 4q. Which is not a odd number.

When r = 1 then

a = 4q+1

Squaring on both sides

a² = [4q+1]²

a²= 16q²+8q+1

a² = 8[2q²+q]+1

a²= 8m+1. Where m = [2q²+q]

Hence similarly u can solve this question by your own

Answer 2

Answer:

Let a be any positive integer

When a is divided by 4, then possible remainders are o,1,2,3.

So when r=0

Then

a = 4q. Which is not a odd number.

When r = 1 then

a = 4q+1

Squaring on both sides

a² = [4q+1]²

a²= 16q²+8q+1

a² = 8[2q²+q]+1

a²= 8m+1. Where m = [2q²+q]