Workbook Wednesdays: Answer

Here is the answer to Wednesday’s Challenge Question from Manhattan GMAT. Check back next week for another Workbook Wednesday question!

Question

If n is an integer and n⁴ 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

Answer

The prime factors of n⁴ are really four sets of the prime factors of the integer n.

Since n⁴ is divisible by 32 (or 2⁵), n⁴ must be divisible by 2 at least 5 times. What does this tell us about the integer n?

If n is divisible by only one 2, then n⁴ would be divisible by exactly four 2′s (since the prime factors of n⁴ have no source other than the integer n).

But we know that n⁴ is divisible by at least five 2′s! This means that n must be divisible by at least two 2′s (which means that n⁴ must be divisible by eight 2′s). Thus, we know that the integer n must be divisible by 4.

Now that we know that n is divisible by 4, we can consider what happens when we divide n by 32.

If we divide n by 32 we can represent this mathematically as follows:

n = 32b + c (where b is the number of times 32 goes into n and c is the integer remainder)

We know that n is divisible by 4 so we can rewrite this as:

4x = 32b + c(where x is an integer)

This equation can be simplified, by dividing both sides by 4 as follows:

x = 8b + c/4

Since we know that x is an integer, the sum of 8b and c/4 must yield an integer. We know that 8b is an integer so c/4 must be also be an integer. Therefore, c, the remainder, must be divisible by 4.

Only answer choice B qualifies. The remainder when n is divided by 32 could be 4. It could not be any of the other answer choices. The correct answer is B.