Math question - confused?

Question

What is the remainder when 3^(4n+ 2) + m is divided by 10

This is a data sufficiency question and one of the prompts is:

m = 1. This is sufficient. Please explain why.

Answer 1

easy
3^(4n+2) = 9^(2n+1) = (10-1)^(2n+1)
(10-1)^(2n+1) mod 10 = -1

so 3^(4n+2) +m mod 10 = m-1 mod 10

so remainder = m-1 mod 10
if m= 1 remainder = 0

Answer 2

hmmm.
n=1 : 3^6 + m ( mod10 ) = m -1
n=2 : 3^(10) + m (mod10) = m + 1
n=3 : 3^14 + m mod(10) = m -1
etc.

yuo cant say the exact raminder since m can be anything

but the remainder is m-1 or m+1 ( when m is reducred mod 10 )

Answer 3

n has to be a number like 6, 10, 14, 18...
Three to these powers be simplified like this.. 9*9*9...

odd powers of nine look like
9 * 9 * 9 = 729
9 * 9 * 9 * 9 * 9 = 59049
and always end in 9.
Adding one to any number that is an odd power of 9 will give you a number divisible by 10.