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# If x, a, and b are positive integers such that when x is

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If x, a, and b are positive integers such that when x is [#permalink]

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29 Sep 2010, 22:27
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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
[Reveal] Spoiler: OA
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29 Sep 2010, 22:49
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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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29 Sep 2010, 23:39
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abhi758 wrote:
Thanks Bunnel!! Do you suggest using a particular strategy for these problems or using different strategy for every problem and whichever fits the bill for the given question..?

What do you mean by "these problems"? Remainder problems or must be true problems?

Anyway...

Remainders:
Theory: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html

Practice PS:
remainder-101074.html
remainder-problem-92629.html
number-properties-question-from-qr-2nd-edition-ps-96030.html
remainder-when-k-96127.html
ps-0-to-50-inclusive-remainder-76984.html
good-problem-90442.html
remainder-of-89470.html
number-system-60282.html
remainder-problem-88102.html

Practice DS:
remainder-problem-101740.html
remainder-101663.html
ds-gcd-of-numbers-101360.html
data-sufficiency-with-remainder-98529.html
sum-of-remainders-99943.html
ds8-93971.html
need-solution-98567.html
gmat-prep-ds-remainder-96366.html
gmat-prep-ds-93364.html
ds-from-gmatprep-96712.html
remainder-problem-divisible-by-86839.html
gmat-prep-2-remainder-86155.html
remainder-94472.html
remainder-problem-84967.html

COULD or MUST be true questions:
ds-number-theory-101025.html?hilit=must%20true
number-properties-question-101150.html?hilit=must%20true
must-be-true-101575.html?hilit=must%20true
gmat-prep-question-101282.html?hilit=must%20true
ab-2-c-is-even-101751.html?hilit=must%20true
mgmat-inequalities-101732.html?hilit=must%20true#p788920
division-and-inequalities-87707.html?hilit=could%20true%20following#p666131

Hope it helps.
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Re: If x, a, and b are positive integers such that when x is div [#permalink]

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04 Sep 2012, 01:14
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sanjoo wrote:
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 divided by A, remainder is B, this implies A > B
When divided by B, remainder is A-2, this implies B > A -2

Combining both,
B < A < (B + 2)
Since, A and B are integers,
A = B + 1

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Re: If x, a, and b are positive integers such that when x is [#permalink]

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03 Nov 2013, 01:50
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This problem can be solved by plugging numbers. For instance if x=5, a=3, b=2 we get:
...when x is divided by a, the remainder is b... > 5/3=1+2/3
...and when x is divided by b, the remainder is a-2... > 5/2=2+1/2

Both work. If we plug numbers into the answers, only D will work.

Please give kudos, if you like this.
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29 Sep 2010, 23:00
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Awesome Bunuel. Was trying hard to figure out the answer. All the information was just readily available but was looking all around.

+1 to you.
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07 Nov 2013, 05:50
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ankit41 wrote:
Bunuel wrote:
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 $$b=a-1$$).

Hi Bunuel,

I solved this problem with a bit different approach
x = p*a + b..........eqn(1)
and x = q*b + (a-2)..............eqn(2)

now, equating eqn(1) and eqn(2)
p*a + b = q*b + (a-2)

a(p-1) = b(q-1) - 2
if we put p = q = 3

we get, 2a = 2b - 2
or a = b - 1
or a + 2 = b + 1 which is option E

would pl tell me where am i wrong with my approach??
Thanks.

You cannot assign arbitrary values to p and q and say that p = q = 3.
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Re: If x, a, and b are positive integers such that when x is [#permalink]

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21 Oct 2014, 18:26
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So the thing to keep in mind with remainder problems like this is that it usually involves finding a value from knowing that the remainder must be smaller than the quotient. From this, we know that a>b and that b>a-2, put these two together and we get: a>b>a-2 and since these are all positive integers then there b must equal a-2
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Re: If x, a, and b are positive integers such that when x is [#permalink]

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13 Nov 2014, 15:20
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Since we know that the remainder is b when x is divided by a we know that b is smaller than a. We also know that a-2 is smaller than b, because a-2 is the remainder when x is divided by b. If we put those two together we know that:
a-2<b<a

Because we know that x, a and b are all positive integers we also know that b must be a-1, because it is an integer that is bigger than a-2 and smaller than a
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29 Sep 2010, 23:06
Thanks Bunnel!! Do you suggest using a particular strategy for these problems or using different strategy for every problem and whichever fits the bill for the given question..?
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29 Sep 2010, 23:18
Simply amazing bunuel...... you are a genius....
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29 Sep 2010, 23:29
cool, BUenel, i had go arround with the formulations but can not find that A-2<B tks much
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30 Sep 2010, 05:57
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Bunuel wrote:
abhi758 wrote:
Thanks Bunnel!! Do you suggest using a particular strategy for these problems or using different strategy for every problem and whichever fits the bill for the given question..?

What do you mean by "these problems"? Remainder problems or must be true problems?

Anyway...

Remainders:
Theory: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html

Practice PS:
remainder-101074.html
remainder-problem-92629.html
number-properties-question-from-qr-2nd-edition-ps-96030.html
remainder-when-k-96127.html
ps-0-to-50-inclusive-remainder-76984.html
good-problem-90442.html
remainder-of-89470.html
number-system-60282.html
remainder-problem-88102.html

Practice DS:
remainder-problem-101740.html
remainder-101663.html
ds-gcd-of-numbers-101360.html
data-sufficiency-with-remainder-98529.html
sum-of-remainders-99943.html
ds8-93971.html
need-solution-98567.html
gmat-prep-ds-remainder-96366.html
gmat-prep-ds-93364.html
ds-from-gmatprep-96712.html
remainder-problem-divisible-by-86839.html
gmat-prep-2-remainder-86155.html
remainder-94472.html
remainder-problem-84967.html

COULD or MUST be true questions:
ds-number-theory-101025.html?hilit=must%20true
number-properties-question-101150.html?hilit=must%20true
must-be-true-101575.html?hilit=must%20true
gmat-prep-question-101282.html?hilit=must%20true
ab-2-c-is-even-101751.html?hilit=must%20true
mgmat-inequalities-101732.html?hilit=must%20true#p788920
division-and-inequalities-87707.html?hilit=could%20true%20following#p666131

Hope it helps.

Priceless info.Ton thanks
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01 Oct 2010, 07:29
Awesome solution Bunuel. +1

Great question Kudos to GMAT club tests
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01 Oct 2010, 10:35
Awesome!!! I didnt get the fastest way, but through number plug-in I was able to figure out the solution. But your solution is so much better!!
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01 Oct 2010, 19:49
nice and efficient !
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02 Oct 2010, 20:02
Bunnel, thanks so much for the compilation! By 'these problems' I meant 'Must be true' questions in which at times you have more than 1 correct answers. My apologies for the lack of clarity their. Your compilation should be enough to practice. Thanks again!
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20 Oct 2010, 04:38
Nice approach. +1 kudos
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Re: If x, a, and b are positive integers such that when x is [#permalink]

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05 Sep 2012, 23:56
Bunuel wrote:
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 $$b=a-1$$).

Indeed very nice explanation, but for me, for the person who is not that strong in quants sometimes difficult to keep all that concepts in my head and i am jumping to different approaches. Whenever i see must be true questions i plug in some numbers and see which answer works, since it is must be true questions any numbers should work equally. For example in this problem: lets says x=5, a=3 then b=2, so check all the answers and we see that only d works, but if there will be two answers that work try different numbers till we get only one. It could be time consuming, but when we are asked simple expressions it is easy to find numbers that work well.

Bunuel, do you think there are any pitfalls that i should be aware of?
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Re: If x, a, and b are positive integers such that when x is [#permalink]

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20 Apr 2013, 12:57
My approach of plug in numbers was certainly not the best approach, in a zest of GMAT I dont know why I am loosing to think simple....this was a simple algebra which I complicated with numbers
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Re: If x, a, and b are positive integers such that when x is   [#permalink] 20 Apr 2013, 12:57

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