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The number m yields a remainder p when divided by 14 and a remainder q

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The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post Updated on: 15 Jul 2019, 01:14
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Question Stats:

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The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m ?

(A) 45
(B) 53
(C) 72
(D) 85
(E) 100


Source: Nova GMAT
Difficulty Level: 650


I'm looking for an algebraic solution to this question. This question is from Nova GMAT Math Bible. I can evaluate the answer choices individually and arrive at the correct answer but I would like to know how to solve this problem without substituting numbers.

Aside from dividing each choice by 7 and 14 and evaluating the remainders with p = q + 7, I tried the following method.

The question states that m = 14*a + p and m = 7*b + q, and that p = q + 7.
Therefore, m = 14*a + q + 7 = 7*b + q
Solving this through, I get b = 2*a + 1

Using a = 0, 1, 2... I get the following values for m:
q + 7, 14 + q + 7 = q + 21, 28 + q + 7 = q + 35, ...

Using b = 1, 3, 5... I get the same results for m since we used b = 2*a + 1:
q + 7, q + 21, q + 35, ... q + 91, q + 105, ...

I know that remainders can only be 0, 1, 2, ... 6 when dividing by 7.

From here on out, I could see that:
45 -> 35 + 10
53 -> 49 + 4
72 -> 63 + 9
85 -> 77 + 8
100 -> 91 + 9

And that's how I would pick 53, again relying on the choices given.

And that's the extent of my progress in analyzing this problem. You can see that I did not get very far in coming up with an elegant solution. Hoping the seasoned GMAT math pros can help me out. Thank you!

Originally posted by jhabib on 02 Aug 2015, 15:39.
Last edited by SajjadAhmad on 15 Jul 2019, 01:14, edited 3 times in total.
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 02 Aug 2015, 17:02
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jhabib wrote:
The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m ?

(A) 45
(B) 53
(C) 72
(D) 85
(E) 100

I'm looking for an algebraic solution to this question. This question is from Nova GMAT Math Bible. I can evaluate the answer choices individually and arrive at the correct answer but I would like to know how to solve this problem without substituting numbers.

Aside from dividing each choice by 7 and 14 and evaluating the remainders with p = q + 7, I tried the following method.

The question states that m = 14*a + p and m = 7*b + q, and that p = q + 7.
Therefore, m = 14*a + q + 7 = 7*b + q
Solving this through, I get b = 2*a + 1

Using a = 0, 1, 2... I get the following values for m:
q + 7, 14 + q + 7 = q + 21, 28 + q + 7 = q + 35, ...

Using b = 1, 3, 5... I get the same results for m since we used b = 2*a + 1:
q + 7, q + 21, q + 35, ... q + 91, q + 105, ...

I know that remainders can only be 0, 1, 2, ... 6 when dividing by 7.

From here on out, I could see that:
45 -> 35 + 10
53 -> 49 + 4
72 -> 63 + 9
85 -> 77 + 8
100 -> 91 + 9


And that's how I would pick 53, again relying on the choices given.

And that's the extent of my progress in analyzing this problem. You can see that I did not get very far in coming up with an elegant solution. Hoping the seasoned GMAT math pros can help me out. Thank you!



A couple of points before I talk about the question:

1. Please format your question properly.
2. Do not add your own analyses or the correct answer or any sort of explanation. This will prevent people who want to solve the question before looking at the OA. If you want to mention your analysis either you can write it in the next post or hide it by putting it as "spoiler" as I have done above.

I am no seasoned GMAT Pro, but will give it a shot. You have done it correctly by finding \(b = 2a+1 = odd\) but the text above in red is incorrect as remainder when a number is divided by 7 can not be > 7.

Once you know that b = 2a+1 = odd, m = odd multiple of 7 + remainder (0-6) . This is the point for eliminating options as shown below:

The choices should be rewritten as:

45 -> 7*6+3 : Reject as it is an even multiple of 7 + rem
53 -> 7*7 + 4 : Keep this as it is an odd multiple of 7 + rem
72 -> 7*10 + 2 : Reject as it is an even multiple of 7 + rem
85 -> 7*12 + 1 : Reject as it is an even multiple of 7 + rem
100 -> 7*14 + 2 : Reject as it is an even multiple of 7 + rem

Only 1 choice is remaining and is thus the correct answer.
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 22 Mar 2016, 16:54
I solved this question in the following way:

Q<7 so Q can be from 0 to 6 and P<14 so P can be from 0 to 13, BUT the constraint is P=Q +7 so this will mean that P can be in the range from 7 to 13.

m=14k + P or m= 14k + 7 to 13 and look at the answeras, place different values for k, B will give 53 which is 14*3 + 10, the other answers are out of the range
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 24 Mar 2016, 11:12
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Hi All,

This question can be solved with some basic arithmetic, 'brute force' and TESTing THE ANSWERS.

We're told that the number M yields a remainder P when divided by 14 and a remainder Q when divided by 7. We're also told that P = Q + 7. We're asked which one of the following COULD be the value of M.

Since the question asks which answer COULD be the value of M, then that means that there's more than one possible answer. As such, we really just have to play around with the answers that are here and when we find a 'match', we can stop working.

Let's TEST Answer A: 45

IF....
M = 45
45/14 = 3r3 so P = 3
45/7 = 6r3 so Q = 3
3 does NOT = 3+7 though, so this is NOT the answer.

Let's TEST Answer B: 53

IF....
M = 53
53/14 = 3r11 so P = 11
53/7 = 7r4 so Q = 4
11 DOES = 4+7, so this MUST be the answer.

Final Answer:

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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 26 Feb 2018, 11:22
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jhabib wrote:
The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m ?

(A) 45
(B) 53
(C) 72
(D) 85
(E) 100


Let’s go through the answer choices because they are not difficult to work with.

(A) 45

45/14 = 3 r 3 and 45/7 = 6 r 3

We see that p = 3 and q = 3, but it’s given that p = q + 7, so A can’t be the answer.

(B) 53

53/14 = 3 r 11 and 53/7 = 7 r 4

We see that p = 11 and q = 4, and p = q + 7, so B is the answer.

(We will leave the readers to verify that C, D and E couldn’t be the answer either.)

Answer: B
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 02 Mar 2018, 16:25
jhabib wrote:
The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m ?

(A) 45
(B) 53
(C) 72
(D) 85
(E) 100


m/7 gives these successive possible values of m: q, q+7, q+14, q+21, q+28, q+35, q+42, q+49
m/14 gives these successive possible values of m: q+7, q+21, q+35, q+49
q+7, q+21, and q+35 won't work, as q>divisor 7
testing q+49=53,
q=4; q+7=11 yes
m=53
B
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 29 Mar 2018, 22:43
jhabib wrote:
The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If p = q + 7, then which one of the following could be the value of m ?

(A) 45
(B) 53
(C) 72
(D) 85
(E) 100


TESTING VALUES IS BEST WAY HERE:

Start with (C)

72/14 = 5 rem 2 (p)

72/7 = 10 rem 2 (q)

p - q = 0, but we need p - q = 7

Go for (B)

53/14 = 3 rem 11

53/7 = 7 rem 4

p - q = 11 - 4 = 7

Hence (B)
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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New post 08 Apr 2018, 07:04
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I solved this question in the following way:

Q<7 so Q can be from 0 to 6 and P<14 so P can be from 0 to 13, BUT the constraint is P=Q +7 so this will mean that P can be in the range from 7 to 13.

m=14k + P or m= 14k + 7 to 13 and look at the answeras, place different values for k, B will give 53 which is 14*3 + 10, the other answers are out of the range
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Re: The number m yields a remainder p when divided by 14 and a remainder q  [#permalink]

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Re: The number m yields a remainder p when divided by 14 and a remainder q   [#permalink] 14 Jul 2019, 05:37
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