Question

show that the square of an odd positive integer is of the form 8q+1 where m is whole number​

Answer 1

Since any odd positive integer n is of the form 4m+1 or 4m+3.

If n=4m+1, then

n

2

=(4m+1)

2

=16m

2

+8m+1=8m(m+1)+1=8q+1 where q=m(m+1)

If n=4m+3, then

n

2

=(4m+3)

2

=16m

2

+24m+9=8(2m

2

+3m+1)+1=8q+1 where q=2m

2

+3m+1

Hence, n

2

is of the form 8q+1.