Video Explanation

Text Explanation

The first step here is to recognize that in order for k to be an integer, n must be a divisor of 462. 

Let's take the initial equation, , and find the prime factorization of the numerator: 

We can see that 462 is divisible by 2, so divide it by 2: 

462 = 2  231 

This is divisible by 3, so we could divide by 3.  Alternately, you might notice

231 = 210 + 21 = (70  3) + (7  3)  = (70 + 7)3 = 3  77 

231 = 3  7  11

and thus

462 = 2  3  7  11

At this point, one way we could solve this question by plugging in answer choices. Let's try answer choice A, which is 4. 

From here, we can cancel a 2 from the numerator and the denominator, and we are left with: 

Here, we cannot do any more cancelling. So we know that k is a fraction—it is not an integer. So thus, A cannot be the answer. 

We could continue this process for all the other answer choices, until we find one that creates an integer value for k. 

Another possible strategy is to look closely at the prime factorization of 462, and ask ourselves whether we can take one or more of these numbers and use them to create any of our answer choices. Notice that 2  11 = 22. This means that if n = 22 (which is answer choice E), we would have: 

Here the 2s and the 11s cancel, leaving us with

So with answer choice D, k is an integer. So (E) must be the right answer.