Explanation

The given information tells you that n can be expressed in the form n = 45k + 18, where k can be any non-negative integer. Consider how the divisors of 45 and 18 may be related to the divisors of n. Every common divisor of 45 and 18 is also a divisor of any sum of multiples of 45 and 18, like 45k + 18. So any common divisor of 45 and 18 is also a divisor of n. Of the answer choices given, only 9 is a common divisor of 45 and 18.
Thus the correct answer is Choice B.
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Sandy
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the 2nd explaination was more helpful then the first..

Divisibility Rule:

DIVIDEND=(DIVISOR*QUOTIENT)+REMAINDER --(1)

Given:
1) Dividend = 'n'.
2)Divisor=45.
3)Remainder=18.
4)Quotient=?

Let's assume the Quotient to be 'a',

Putting the above values in the formula (1), we get

n=(45*a)+18.
n=45a+18.
n=9(5a+2).

Thus, n must be a multiple of 9.


(or)

When it comes to remainders, we have a nice rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Here, we are told that n divided by 45 leaves a remainder of 18, so the possible values of n are: 18, 63, 108,... etc.

IMPORTANT: the question asks, "Which of the following must be a divisor of n?
So, let's test the smallest possible value of n, which is 18, and check the answer choices.

18 is NOT divisible by 11, 7 or 4, so we can ELIMINATE A, C and E.
So, the correct answer is either B or D

Now test the next possible value of n, which is 63.
63 is NOT divisible by 6, so we can ELIMINATE D

So, by the process of elimination, the correct answer is B.

If k is the greatest positive integer such that 3^k is a divisor of 15! then k =

A. 3
B. 4
C. 5
D. 6
E. 7

I think I'm being exceptionally stupid but I can't get my head around this as 63/ 9 = 7, there is no remainder? Surely a divisor of n shouldn't also be a divisor of 18 otherwise there would be no remainder?

This one didn't really click with me until I tried one of the wrong answers. Let's try 11.

N = 45Q + 18

N = \(\frac{45Q + 18}{11}\)

N = \(\frac{45Q}{11} + \frac{18}{11}\)

You can clearly see now that N can not be an integer after division, which is what we want when looking for a divisor of N, unless BOTH 45Q and 18 are divisible by the answer choice.

So, 11 wouldn't work.

We can then try 9. 9 works. N will still be an integer if it is divided by 9. So, 9 is our answer.


To illustrate:

N = \(\frac{45Q}{9} + \frac{18}{9}\)

N = 5Q + 2

. we can see N will be an integer if we divide by 9. So, 9 works.