Question

If n is an odd integer show that n square minus 1 is divisible by 8.

Answer 1

any +ve integer is of the form 4q+1, 4q+3.
given that "n" is an odd +ve integer
so, by applying Euclid's division lemma, we get,
n=4q+1
n²-1=(4q+1)²-1
( since, n²=(4q+1)², so, n²-1=(4q+1)²-1)
n²-1 = (16q²+8q+1) -1
= 16q²+8q
= 8(2q²+q)
let, 2q²+q be m, where "m" is some +ve integer
so, n²-1 = 8m
now, similarly n=4q+3
n²-1 = (4q+3)²-1
= 16q²+24q+9-1
= 16q²+24q+8
= 8(2q²+3q+1)
= 8m
let 2q²+3q+1 be "m", where "m" is some +ve integer
so, n²-1 = 8m
hence, n²-1 is divisible by 8

MARK ME AS THE BRAINLIEST