Question

show that n square minus 1 is divisible by 8 if n is an positive integer​

Answer 1

N square-1

n is odd positive integer, a=4q+r

r=0,1,2,3

a=4q,4q+1,4q+2,4q+3

Now, from this we cam conclude that 4q+1and 4q+3 are odd positive integer therefore,

Case1=4q+1

=n=4q+1

n square-1=(4q) square+(1)Square +2×4q-1

=16q square+8q

=8q(2q+1)

Case 2=4q+3

=n=4q+3

n Square - 1=(4q+3)Square - 1

=16q Square +9+2×4q×3-1

=16q Square +9+24q-1

=16q Square +24q+8

=8(2q square +3q+1)

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