JMO division

If 8ⁿ is divided by 6, then for n € N, & n is odd, and for n € N,& n is even what is the Remainder?

Answer:

8ⁿ  = (6 + 2)ⁿ = 6ⁿ + n6^n-1.2 +……….n.6.2^n-1 + 2ⁿ

When 8ⁿ is divided by 6, all the terms except the last, i.e. 2ⁿ will get completely divided.

For,

n=1	R=2
n=2	R=4
n=3	R=2
n=4	R=4
n=5	R=2
n=6	R=4

when n is odd number, R = 2

When n is even number, R = 4