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If K is a positive integer such that the remainder when 17 is divided

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If K is a positive integer such that the remainder when 17 is divided  [#permalink]

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New post 26 Apr 2018, 13:34
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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

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Re: If K is a positive integer such that the remainder when 17 is divided  [#permalink]

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New post 26 Apr 2018, 13:48
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


When 17 is divided by K = 15(3*5), the remainder is 2.

The remainder is also 2 when K takes the value 3 or 5.

Therefore, the sum of all possible values of K is \(3+5+15 =\) 23(Option D)
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Re: If K is a positive integer such that the remainder when 17 is divided  [#permalink]

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New post 26 Apr 2018, 13:55
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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Re: If K is a positive integer such that the remainder when 17 is divided  [#permalink]

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New post 27 Apr 2018, 00:11
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


Ans: D

We are given a relation which is 17 = KI + 2 ; I and K are int Positive
so we can write it in this form K = 15/I ; as already given I and K are int I will take the values of Int which divides 15 and leaves no reminder thus those value of K are 3,5,and 15
Sum= 23

Ans D
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Re: If K is a positive integer such that the remainder when 17 is divided  [#permalink]

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New post 30 Apr 2018, 16:32
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 can create the equation:

17/K = Q + 2/K

17 = KQ + 2

15 = KQ

Thus, we see that K is a factor of 15. The factors of 15 are 1, 3, 5, and 15. However, K can’t be 1 since 17 is divisible by 1 (with remainder 0). On the other hand, when K is 3, 5 or 15, we do have a remainder 2 when 17 is divided by K. Thus, the sum of all possible values of K is 3 + 5 + 15 = 23.

Answer: D
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Re: If K is a positive integer such that the remainder when 17 is divided  [#permalink]

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New post 30 Apr 2018, 21:49
17/k gives us remainder 2. So we know that k can be divided by 15. Additionally, with prime factorization, we know 15 is broken down to 5 and 3. 15 + 3 + 5 are the only combinations for this answer and the question asks to sum possible values; therefore, the answer is 23. Answer Choice D.
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Re: If K is a positive integer such that the remainder when 17 is divided &nbs [#permalink] 30 Apr 2018, 21:49
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