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Hi, there. I'm happy to help with this.

First of all, I will say --- the GMAT does ask about divisibility, but this question is much harder than anything I have ever seen the GMAT ask.

Prompt: When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?

This means, for some quotient Q, x/d = Q + 24, or in other words: x = Qd + 24

Statement #1: When 2x is divided by d, the remainder is 23.

x = Qd + 24, so 2x = 2*(Qd + 24) = 2Qd + 48

When we divide 2x = 2Qd + 48 by d, d clearly goes evenly into 2Qd. When we divide 48/d, we get a remainder of 23. That means d must go evenly into 48 - 23 = 25. We see that d can't be 5, because when we divide 48 by 5, the remainder is 3, not 23. The only possibility is d = 25. Statement #1, by itself, is sufficient to determine a definitive value for d.

Statement #2: When 3x is divided by d, the remainder is 22.

x = Qd + 24, so 3x = 3*(Qd + 24) = 3Qd + 72

When we divide 3x = 3Qd + 72 by d, d clearly goes evenly into 3Qd. When we divide 72/d, we get a remainder of 22. That means d must go evenly into 72 - 22 = 50. We see that d can't be 5 or 10, because d must be bigger than the remainder (the divisor is always greater than the remainder). Here, d could be 25 or 50. Both 25 and 50 are bigger than the remainder 22, and both of them, when divided into 72, give a remainder of 22. This statements, by itself, does not determine a unique and definitive value of d --- it leaves open two possibilities. Statement #2 is insufficient.

Answer = A

Again, this problem is a bit trickier than what I've seen the GMAT ask about divisibility & remainders. Here's a somewhat more GMAT-like question on this topic:

http://gmat.magoosh.com/questions/873

When you submit an answer to this question, the next page will give a complete video explanation.

Does all this make sense? Please let me know if you have any further questions about this.

Mike

_________________
Magoosh GMAT Instructor
http://gmat.magoosh.com/

I received a PM asking me to comment.

For any divisor d, the greatest possible remainder is d-1.
Since the remainder in the question stem is 24:
24 ≤ d-1
d ≥ 25.

When a positive integer 'x' is divided by a divisor 'd', the remainder is 24:
In other words, x is 24 more than a multiple of d:
x = kd + 24.

Statement 1: When 2x is divided by d, the remainder is 23.
In other words, 2x is 23 more than a multiple of d:
2x = md + 23.
Since in the question stem x = kd + 24, it must also be true that 2x = (2k)d + 48.
Equating the two representations of 2x:
md + 23 = (2k)d + 48
md - (2k)d = 25.
(m-2k)d = 25.
Since d must be factor of 25, and d≥25, d=25.
SUFFICIENT.

Statement 2: When 3x is divided by d, the remainder is 22.
In other words, 3x is 22 more than a multiple of d:
3x = md + 22.
Since in the question stem x = kd + 24, it must also be true that 3x = (3k)d + 72.
Equating the two representations of 3x:
md + 22 = (3k)d + 72
md - (3k)d = 50.
(m-3k)d = 50.
Since d must be factor of 50, and d≥25, either d=25 or d=50.
INSUFFICIENT.

The correct answer is A.

_________________
Mitch Hunt
GMAT Private Tutor and Instructor
GMATGuruNY@gmail.com
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