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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## The number m yields a remainder p when divided by 14 . . . . tagged by: Vincen ##### This topic has 4 expert replies and 1 member reply ## The number m yields a remainder p when divided by 14 . . . . 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 The OA is B. How can I choose the correct answer? Is there an easy and fast way to solve this PS question? I'd be thankful. ### GMAT/MBA Expert Legendary Member Joined 14 Jan 2015 Posted: 2667 messages Followed by: 122 members Upvotes: 1153 GMAT Score: 770 Vincen 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 The OA is B. How can I choose the correct answer? Is there an easy and fast way to solve this PS question? I'd be thankful. We could back-solve. A) m = 45. 45/14 = 3 + 3/14, so the remainder, p, would be 3. Clearly this is out, as we're told p is 7 more than q, and thus p has to be at least 7. B)m = 53. 53/14 = 3 + 11/14, so the remainder, p, would be 11. Now we'll find q. 53/7 = 7 + 4/7, giving us a remainder, or q, of 4. Well, 11 is 7 more than 4, so our condition of p = q + 7 is met, We've got our answer: B _________________ Veritas Prep | GMAT Instructor Veritas Prep Reviews Save$100 off any live Veritas Prep GMAT Course

Last edited by DavidG@VeritasPrep on Tue Jan 16, 2018 2:11 pm; edited 1 time in total

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### GMAT/MBA Expert

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Vincen 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
Like David, I would solve by PLUGGING IN THE ANSWERS.
That said, here's an algebraic approach:

When dividing by 7, the remainder can be any integer between 0 and 6, inclusive.
Thus, 0â‰¤qâ‰¤6.

The number m yields a remainder q when divided by 7.
Thus, m is a MULTIPLE OF 7 plus q:
m =7a + q.

The number m yields a remainder p when divided by 14.
Thus, m is a MULTIPLE OF 14 plus p:
m = 14b + p.

The prompt indicates that p = q+7.
Substituting p = q+7 into m = 14b+p, we get:
m = 14b + q + 7.

Since the expressions in blue are both equal to m, they must be equal to each other:
7a + q = 14b + q + 7
7a = 14b + 7
a = 2b + 1.

The resulting equation implies that a can be ANY ODD INTEGER.
Since a must be odd, and m = 7a + q, we get the following options for m:
m = (7*1) + q = 7 + q
m = (7*3) + q = 21 + q
m = (7*5) + q = 35 + q
m = (7*7) + q = 49 + q
And so on.

Since 0â‰¤qâ‰¤6, the following ranges for m are possible:
m = 7+q --> m is between 7 and 13, inclusive
m = 21+q --> m is between 21 and 27, inclusive
m = 35+q --> m is between 35 and 41, inclusive
m = 49+q --> m is between 49 and 55, inclusive.

Since 53 is within the range in green, option B is a possible value for m.

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Hi Vincen,

We're told that the number M yields a remainder of P when divided by 14 and a remainder Q when divided by 7. If P = Q + 7, then we're asked which one of the following COULD be the value of M. This question can be solved in a number of different ways. Here's how you can use some Number Properties - and the answer choices - to find the correct answer.

To start, since P = Q+7 (and the two variables refer to the REMAINDERS), then P MUST be at least 7. We can now divide each of the answer choices by 14 and check the remainders. Any answer that has a remainder LESS than 7 can be eliminated....

Answer A: 45.... 45/14 = 3 r 3. Eliminate A
Answer B: 53.... 53/14 = 3 r 11
Answer C: 72... 72/14 = 5 r 2. Eliminate C
Answer D: 85... 85/14 = 6 r 1. Eliminate D
Answer E: 100... 100/14 = 7 r 2. Eliminate E

There's only one answer that 'fits' this piece of information:

GMAT assassins aren't born, they're made,
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This is an interesting question because 53 is just one of a number of different valid answers. Therefore trying algebra could get a person tied in knots!
A super-short method is as follows:

(We will ignore the remainders for most of this solution!)
m= 14a + p = 14a + 7 + q
m = 7b + q
14a is EVEN
And 7b = 14a + 7 so 7b is ODD, hence b is ODD too.
Hence, (m-q) is an ODD multiple of 7, where q<7
Also m > 14
Therefore, m has values in the ranges 21-27, 35-41, 49-55, 63-69, 77-83, 91-97, etcâ€¦
Only Answer B (53) fits into one of these valid ranges.

### GMAT/MBA Expert

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Vincen 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
Letâ€™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 itâ€™s given that p = q + 7, so A canâ€™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 couldnâ€™t be the answer either.)

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