Problem Solving

If n is an integer and n^4 is divisible by 32, which of the following could be the remainder when n is divided by 32?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

First I determined which integers when raised to the 4th are divisible by 32.
32 = 2^5 so n^4 must have 2^5 as a factor

1^4 no
2^4 no
3^3 no
4^4 = 2^8 yes
5^4 no
6^4 no
7^4 no
8^4 = 2^12 yes

following the same pattern n is a multiple of 4

Trying out n=4: 4 divided by 32 is zero remainder 4
So 4 is an option. (Shortcut I used)

To check: Since n is a multiple of 4 try out all multiples of 4 until the pattern is revealed.

8 divided by 32 is zero remainder 8
12 divided by 32 is zero remainder 12
16 divided by 32 is zero remainder 16
20 divided by 32 is zero remainder 20
24 divided by 32 is zero remainder 24
28 divided by 32 is zero remainder 28
32 divided by 32 is 1 remainder 0
36 divided by 32 is 1 remainder 4
40 divided by 32 is 1 remainder 8

and the cycle continues.

The remainders (8,12,16,20,24,28,0,4,8 ) are possible options but not all of them are answer choices. The only value that is an answer choice is 4. So B is the answer.

Although the explanation may seem long, the computations are really short and the check can be done in your head.