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# If (x # y) represents the remainder that results when the po

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Joined: 18 Oct 2013
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If (x # y) represents the remainder that results when the po  [#permalink]

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Updated on: 29 Mar 2014, 14:55
21
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Difficulty:

35% (medium)

Question Stats:

70% (01:37) correct 30% (01:49) wrong based on 278 sessions

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If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

A. 8
B. 9
C. 16
D. 23
E. 24

Is there a quick and easy way to find divisors to an integer/expression when we are looking for a distinct remainders?

23
, I found the answer by calculating and doing several divisions, is there an easier way?

Originally posted by jokkemauritzen on 29 Mar 2014, 14:23.
Last edited by Bunuel on 29 Mar 2014, 14:55, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Joined: 02 Sep 2009
Posts: 55681
Re: If (x # y) represents the remainder that results when the po  [#permalink]

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29 Mar 2014, 15:02
2
5
jokkemauritzen wrote:
If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

A. 8
B. 9
C. 16
D. 23
E. 24

Is there a quick and easy way to find divisors to an integer/expression when we are looking for a distinct remainders?

23
, I found the answer by calculating and doing several divisions, is there an easier way?

(x # y) represents the remainder that results when the positive integer x is divided by the positive integer y.

Thus (16 # y) = 1 implies that $$16=yq+1$$ --> $$15=yq$$ --> y is a factor of 15. The factors of 15 are 1, 3, 5, and 15. Now, y cannot be 1, since 16 divided by 1 yields the remainder of 0 not 1.

Therefore the sum of all the possible values of y is 3+5+15=23.

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Joined: 12 Sep 2015
Posts: 3783
Re: If (x # y) represents the remainder that results when the po  [#permalink]

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19 Apr 2019, 06:07
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Top Contributor
jokkemauritzen wrote:
If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

A. 8
B. 9
C. 16
D. 23
E. 24

If y > 16, (16 # y) = 16, so we need only check the values from 1 to 15
Also, we need not check the FACTORS of 16, since they will all yield a remainder of 0
We're left with:
(16 # 3) = 1 KEEP!
(16 # 5) = 1 KEEP
(16 # 7) = 2
(16 # 9) = 7
(16 # 10) = 6
(16 # 11) = 5
(16 # 12) = 4
(16 # 13) = 3
(16 # 14) = 2
(16 # 15) = 1 KEEP

3 + 5 + 15 = 23

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Re: If (x # y) represents the remainder that results when the po  [#permalink]

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26 Apr 2019, 15:39
jokkemauritzen wrote:
If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1?

A. 8
B. 9
C. 16
D. 23
E. 24

16/15 has a remainder of 1.

16/5 has a remainder of 1.

16/3 has a remainder of 1.

So the sum of all possible values of y is 15 + 5 + 3 = 23.

Alternate Solution:

We are looking for all values of y such that 16 divided by y produces a remainder of 1. Then, 16 - 1 = 15 must be divisible by y. Excluding y = 1 (which produces a remainder of 0); the possibilities for y are 15, 5 and 3. Thus, the sum of all possible values of y is 15 + 5 + 3 = 23.

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Re: If (x # y) represents the remainder that results when the po   [#permalink] 26 Apr 2019, 15:39
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