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

Question

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!

ENGRTOMBA2018 · Answer

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.

kzivrev · Answer

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

EMPOWERgmatRichC · Answer

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:  B

GMAT assassins aren't born, they're made,
Rich

JeffTargetTestPrep · Answer

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

gracie · Answer

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

AkshdeepS · Answer

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)

SidKan · Answer

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

BrushMyQuant · Answer

Given that The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. And we need to find If  p = q + 7 , what can be the value of m

Theory: Dividend = Divisor*Quotient + Remainder

Now, m when divided by 14 gives p remainder and p = q + 7

m -> Dividend
14 -> Divisor
a -> Quotient (Assume)
q + 7 -> Remainders
=> m = 14a + q + 7 = 7*2a + 7 + q = 7*(2a + 1) + q = odd multiple of 7 + q (where q is less than 7)

let's take each option choice and check which one satisfies this

(A) 45  = 42 + 3 = Even multiple of 7 + 3 =>   NOT POSSIBLE

(B) 53  = 49 + 4 = Odd multiple of 7 + 4 =>   POSSIBLE  
In Test we don't need to check further but I am solving to complete the solution

(C) 72  = 70 + 2 = Even multiple of 7 + 2 =>   NOT POSSIBLE

(D) 85  = 84 + 1 = Even multiple of 7 + 1 =>   NOT POSSIBLE

(E) 100  = 98 + 2 = Even multiple of 7 + 2 =>   NOT POSSIBLE

So,  Answer will be B 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

bumpbot · Answer

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

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