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The number m yields a remainder p when divided by 14 . . . .

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The number m yields a remainder p when divided by 14 . . . .

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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

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.

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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

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

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

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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, 0≤q≤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 0≤q≤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.

The correct answer is B.

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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:

Final Answer: B

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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, etc…
Only Answer B (53) fits into one of these valid ranges.

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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
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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