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## Divisor Test - Gmat-Math Blogspot

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amitkoul777@gmail.com Just gettin' started!
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Divisor Test - Gmat-Math Blogspot Sun Mar 11, 2012 10:48 am
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I found this problem at http://gmat-math.blogspot.com/ but they have not provided any answer to it.
Can anyone help in solving this please ?

When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?
1. When 2x is divided by d, the remainder is 23.
2. When 3x is divided by d, the remainder is 22.

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gopinathhyd Just gettin' started!
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Sun Mar 11, 2012 1:37 pm
since we are given 1 equation, from which we can get dq + 24 = x (imagine q as quotient)any way Q is common in all our reactions.

St.1 gives us one more equation dQ + 23 = 2X (using this equation we can calulate X = -1. so we can find d. SUFFICIENT

St.2 Gives another similar equation . Sufficient.

IMO: D

I hope the answer is right.

amitkoul777@gmail.com Just gettin' started!
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Sun Mar 11, 2012 4:02 pm
How can u assume that Q is same in all reactions ?
Take an example of d=25 and x=124. When you divide 124 by 25, the remainder is 24 and Q1=4.

Now from statement 1, when you double x, it becomes 248. Dividing 248 by 25 gives remainder as 23 but Q2=9. So Q1 Not equal Q2.

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Mike@Magoosh GMAT Instructor
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Tue Mar 13, 2012 9:06 am
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.

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.

Mike

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

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GMATGuruNY GMAT Instructor
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Tue Mar 13, 2012 1:44 pm
amitkoul777@gmail.com wrote:
I found this problem at http://gmat-math.blogspot.com/ but they have not provided any answer to it.
Can anyone help in solving this please ?

When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?
1. When 2x is divided by d, the remainder is 23.
2. When 3x is divided by d, the remainder is 22.

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.

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akshay1006 Just gettin' started!
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Tue Mar 20, 2012 8:45 am
hi i got help from http://gmatselfstudybooster.blogspot.in/ try it out

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