Wording and answer of algebra for GRE

Question

I am studying for GRE and have been stumped by

1. The question

2. The answer

Question:

When the positive integer n is divided by 45, the remainder is 18. Which of the following must be a divisor of n?

Choices

a) 11

b) 9

c) 7

d) 6

e) 4

Are they asking what is k, where n/45 = k ? Because I think divisor is something that divides n, i.e. n/k = 45

But then their explanation is more confusing

Explanation

The given information tells you that n can be expressed in the form n = 45k + 18, where k can be any nonnegative integer.

Consider how the divisors of 45 and 18 may be related to the divisors of n.

But I am not understanding what this has to do with k.

Every common divisor of 45 and 18 is also a divisor of any sum of multiples of 45 and 18, like 45k + 18.

But what does this have to do with k?

So any common divisor of 45 and 18 is also a divisor of n. Of all the answer choices given, only 9 is a common divisor of 45 and 18. Thus the correct answer is Choice B.

Why must there be a common divisor o 45 and 18. I thought 18 was the remainder of n/45 = k.

So are they saying k = 9? I am really really confused. I tried asking someone who is good at math, and he couldn't figure it out.

Answer 1

The are asking for the divisor. What can $n$ be divided by? As $n = 45k + 18 = 9(5k+2)$, what do you think $n$ can be divided by?

Answer 2

You know that $n=45k+18$, for some $k$. Thus, immediately, $9$ is a divisor of $n$, as it divides both $45$ and $18$.

Can we dismiss all other cases?

(a) $n=18$ is not divisible by $11$
(c) $n=18$ is not divisible by $7$
(d) $n=63$ is not divisible by $6$
(e) $n=18$ is not divisible by $4$