Problem Solving

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.

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

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.

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.