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# M16 Q35 PS from GMAT club tests

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M16 Q35 PS from GMAT club tests [#permalink]  14 Aug 2008, 08:25
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$$X$$ , $$A$$ , and $$B$$ are positive integers. When $$X$$ is divided by $$A$$ , the remainder is $$B$$ . If when $$X$$ is divided by $$B$$ , the remainder is $$A - 2$$ , which of the following must be true?

(A) $$A$$ is even
(B) $$X + B$$ is divisible by $$A$$
(C) $$X - 1$$ is divisible by $$A$$
(D) $$B = A - 1$$
(E) $$A + 2 = B + 1$$

Source: GMAT Club Tests - hardest GMAT questions

REVISED VERSION OF THIS QUESTION IS HERE: m16-q35-ps-from-gmat-club-tests-68861.html#p1220163
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Re: PS from GMAT club test M16 Q35 [#permalink]  14 Aug 2008, 08:58
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durgesh79 wrote:
spent 10 minutes on this one .... and finally guessed it and it was right ...

X, A, and B are positive integers. When X is divided by A, the remainder is B. When X is divided by B, the remainder is A-2 . Which of the following must be true?

A is even
X+B is divisible by A
X-1 is divisible by A
B = A-1
A+2 = B+1

D firm:

When X is divided by A, the remainder is B.
-- means A>B (remainder always less A)

e.g. 5/10 -- remainder -- 5
25/10 -- remainder - 5 ( B<A)

When X is divided by B, the remainder is A-2
that means B>A-2 --> B+2>A

Combine both equations.

B+2>A>B (A, B are positive intergers..
there is only one values posible between B and B+2.. that is B+1
so A=B+1
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Last edited by x2suresh on 14 Aug 2008, 09:01, edited 1 time in total.
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Re: PS from GMAT club test M16 Q35 [#permalink]  14 Aug 2008, 08:42
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durgesh79 wrote:
spent 10 minutes on this one .... and finally guessed it and it was right ...

X, A, and B are positive integers. When X is divided by A, the remainder is B. When X is divided by B, the remainder is A-2 . Which of the following must be true?

A is even
X+B is divisible by A
X-1 is divisible by A
B = A-1
A+2 = B+1

D guessing..

From
X / A = p * A + B
X/B = Q* B + (A-2)

So.. B < A and (A-2) < B

So B < A < B+2

Since A and b ARE INTEGERS

A should be B+1
So I vote for D

spent more than 3 Mins
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Re: M16 Q35 PS from GMAT club tests [#permalink]  03 May 2013, 04:18
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Expert's post
durgesh79 wrote:
$$X$$ , $$A$$ , and $$B$$ are positive integers. When $$X$$ is divided by $$A$$ , the remainder is $$B$$ . If when $$X$$ is divided by $$B$$ , the remainder is $$A - 2$$ , which of the following must be true?

(A) $$A$$ is even
(B) $$X + B$$ is divisible by $$A$$
(C) $$X - 1$$ is divisible by $$A$$
(D) $$B = A - 1$$
(E) $$A + 2 = B + 1$$

Source: GMAT Club Tests - hardest GMAT questions

spent 10 minutes on this one .... and finally guessed it and it was right ...

BELOW IS REVISED VERSION OF THIS QUESTION:

If $$x$$, $$a$$, and $$b$$ are positive integers such that when $$x$$ is divided by $$a$$, the remainder is $$b$$ and when $$x$$ is divided by $$b$$, the remainder is $$a-2$$, then which of the following must be true?

A. $$a$$ is even
B. $$x+b$$ is divisible by $$a$$
C. $$x-1$$ is divisible by $$a$$
D. $$b=a-1$$
E. $$a+2=b+1$$

When $$x$$ is divided by $$a$$, the remainder is $$b$$ --> $$x=aq+b$$ --> $$remainder=b<a=divisor$$ (remainder must be less than divisor);
When $$x$$ is divided by $$b$$, the remainder is $$a-2$$ --> $$x=bp+(a-2)$$ --> $$remainder=(a-2)<b=divisor$$.

So we have that: $$a-2<b<a$$, as $$a$$ and $$b$$ are integers, then it must be true that $$b=a-1$$ (there is only one integer between $$a-2$$ and $$a$$, which is $$a-1$$ and we are told that this integer is $$b$$, hence $$b=a-1$$).

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Re: M16 Q35 PS from GMAT club tests [#permalink]  28 Apr 2011, 09:20
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spent more than 10 min trying to solve, finally substituted numbers (A=3, B=2) to arrive at (D) ... great approach by x2suresh ... divisor>remainder ... life can be so simple
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Re: M16 Q35 PS from GMAT club tests [#permalink]  11 May 2013, 06:27
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Simple Plugin type Question :-

X = 11
A = 3
B = 2
...........................

Substitute in answer choices and you will find that only D satisfies the condition.
Solved in under 1:30 min.
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Re: PS from GMAT club test M16 Q35 [#permalink]  14 Aug 2008, 08:53
you never define p

for this:
Quote:
X / A = p * A + B

to be true, p must be negative. that doesn't make any sense.

$$\frac{x}{A} = Ap + B$$

$$x = pA^2 + AB$$

Take x = 9, a = 7

$$\frac{9}{7} = 7p + 9$$

$$9 = 49p + 63$$

$$-54 = 49p$$

$$-\frac{54}{49} = p$$

Do you mean this:

$$\frac{x}{a} = \frac{x-b}{a} + \frac{b}{a}$$
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Re: PS from GMAT club test M16 Q35 [#permalink]  14 Aug 2008, 09:01
chan4312 wrote:
durgesh79 wrote:
D guessing..

From
X / A = p * A + B
X/B = Q* B + (A-2)

spent more than 3 Mins

Above equations are incorrect.

X = p * A + B
X = Q* B + (A-2)
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Re: M16 Q35 PS from GMAT club tests [#permalink]  27 Apr 2010, 22:07
Also from substitution method it we can see that by taking numbers for A and B , taking 3 and 2 the closer number would be 5, as the conditions get satisfied so we can assure that 3 and 2 are A,B. Thus satisfies the conditions.

Took 2 min
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Re: M16 Q35 PS from GMAT club tests [#permalink]  28 Apr 2011, 14:57
D.

Plugging in numbers and backsolving - Solving from the last option.

X=5;A=3;B=2 -> We can eliminate all the answer choices except D. <1 Min
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Re: M16 Q35 PS from GMAT club tests [#permalink]  30 Jun 2011, 00:33
I know this is a dumb question- but I cant solve these 2 inequalities mathematically :

b < a
a-2 <b
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Re: M16 Q35 PS from GMAT club tests [#permalink]  02 May 2012, 10:29
Excellent Concept Problem: D is correct answer.

When X is divided by A, the remainder is B. => A > B
When X is divided by B, the remainder is (A-2). => B > A - 2
Combining the two inequalities: A > B > A - 2
i.e.: B is between A and (A-2), so B must be equal to A-1.
B = A - 1. (D) is the correct answer.
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Re: PS from GMAT club test M16 Q35 [#permalink]  05 May 2013, 12:47
x2suresh wrote:

X / A = p * A + B
X/B = Q* B + (A-2)

Above equations are incorrect.

X = p * A + B
X = Q* B + (A-2)

took me almost 10 minutes to accept that the first 2 equations are really incorrect. But it helps to substitude numbers.
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Re: PS from GMAT club test M16 Q35 [#permalink]  09 May 2013, 17:01
Ultimor wrote:
x2suresh wrote:

X / A = p * A + B
X/B = Q* B + (A-2)

Above equations are incorrect.

X = p * A + B
X = Q* B + (A-2)

took me almost 10 minutes to accept that the first 2 equations are really incorrect. But it helps to substitude numbers.

Agreed. It's a good trap, one that testmakers know people fall into, easily.

[old way of thinking & bad habits!!!]
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Re: M16 Q35 PS from GMAT club tests [#permalink]  18 Apr 2014, 05:50
Bunuel wrote:
durgesh79 wrote:
$$X$$ , $$A$$ , and $$B$$ are positive integers. When $$X$$ is divided by $$A$$ , the remainder is $$B$$ . If when $$X$$ is divided by $$B$$ , the remainder is $$A - 2$$ , which of the following must be true?

(A) $$A$$ is even
(B) $$X + B$$ is divisible by $$A$$
(C) $$X - 1$$ is divisible by $$A$$
(D) $$B = A - 1$$
(E) $$A + 2 = B + 1$$

Source: GMAT Club Tests - hardest GMAT questions

spent 10 minutes on this one .... and finally guessed it and it was right ...

BELOW IS REVISED VERSION OF THIS QUESTION:

If $$x$$, $$a$$, and $$b$$ are positive integers such that when $$x$$ is divided by $$a$$, the remainder is $$b$$ and when $$x$$ is divided by $$b$$, the remainder is $$a-2$$, then which of the following must be true?

A. $$a$$ is even
B. $$x+b$$ is divisible by $$a$$
C. $$x-1$$ is divisible by $$a$$
D. $$b=a-1$$
E. $$a+2=b+1$$

When $$x$$ is divided by $$a$$, the remainder is $$b$$ --> $$x=aq+b$$ --> $$remainder=b<a=divisor$$ (remainder must be less than divisor);
When $$x$$ is divided by $$b$$, the remainder is $$a-2$$ --> $$x=bp+(a-2)$$ --> $$remainder=(a-2)<b=divisor$$.

So we have that: $$a-2<b<a$$, as $$a$$ and $$b$$ are integers, then it must be true that $$b=a-1$$ (there is only one integer between $$a-2$$ and $$a$$, which is $$a-1$$ and we are told that this integer is $$b$$, hence $$b=a-1$$).

Do you have any suggestions on how one can be quick at finding this insight? (And finding the correct insight quickly in general?)
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Re: M16 Q35 PS from GMAT club tests [#permalink]  18 Apr 2014, 06:58
but how can we eliminate others. for example if we start from option A how can be eliminate this option???
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Re: M16 Q35 PS from GMAT club tests [#permalink]  12 May 2014, 22:55
1st statement implies, A > B ; as B is reminder while A is divisor
2nd statement implies B > A-2 ; as A-2 is reminder while B is divisor

So, together of them implies - B+2>A>B => B = A+1
Re: M16 Q35 PS from GMAT club tests   [#permalink] 12 May 2014, 22:55
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