Bunuel wrote:

If K is a positive integer such that the remainder when 17 is divided by K is 2, what is the sum of all the possible values of K?

(A) 8

(B) 18

(C) 20

(D) 23

(E) 25

We'll show two approaches:

If it isn't immediately clear where to look for an abstract solution, we'll list our options.

This is an Alternative approach.

1*17=17 so remainder 0

2*8=16 --> remainder 1

3*5 = 15 --> remainder 2 so good!

4*4 = 16 --> no good

5*3 = 15 --> good!

6*2 = 12 --> no good

7*2 = 14 --> no good

8*2 = 16 --> no good.

9,10... these are too large as multiplying by 2 makes them larger than 17.

Only 15 works as 15*1 = 15

Then our sum is 3 + 5 +15 = 23.

(D) is our answer.

A shorter, more abstract solution:

Questions dealing with remainders can often be solved with number properties and very few calculations.

We'll look for such a solution, a Logical approach.

If 17 divided by k gives a remainder of 2, then k must divide 17 - 2 = 15.

That is, k is either 1,3,5 or 15. But k=1 can't work as 17 divided by 1 has remainder 0.

So the only options are 3,5,15 and our sum is 23.

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