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 + rem72 -> 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.