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# When the positive integer A is divided by 5 and 7, the

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Joined: 29 Sep 2008
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When the positive integer A is divided by 5 and 7, the [#permalink]

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08 Nov 2010, 08:59
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When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30
[Reveal] Spoiler: OA

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Joined: 02 Sep 2009
Posts: 43312

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08 Nov 2010, 09:17
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mrinal2100 wrote:
When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

i used the numbers and reached at two numbers 18 and 53 and 53-18 gives 35.is there any better way to solve this question

When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively: $$A=5q+3$$ (A could be 3, 8, 13, 18, 23, ...) and $$A=7p+4$$ (A could be 4, 11, 18, 25, ...).

There is a way to derive general formula based on above two statements:

Divisor will be the least common multiple of above two divisors 5 and 7, hence $$35$$.

Remainder will be the first common integer in above two patterns, hence $$18$$ --> so, to satisfy both this conditions A must be of a type $$A=35m+18$$ (18, 53, 88, ...);

The same for B (as the same info is given about B): $$B=35n+18$$;

$$A-B=(35m+18)-(35n+18)=35(m-n)$$ --> thus A-B must be a multiple of 35.

manhattan-remainder-problem-93752.html?hilit=derive#p721341
good-problem-90442.html?hilit=derive#p722552

Hope it helps.
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08 Nov 2010, 10:07
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mrinal2100 wrote:
When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

i used the numbers and reached at two numbers 18 and 53 and 53-18 gives 35.is there any better way to solve this question

If I have a number n which when divided by 5 gives a remainder 3 and when divided by 7 gives a remainder 4, the number is of the form:
n = 5a + 3
n = 7b + 4
I will need to check for the smallest such number.
I put b = 1. n = 11. Is it of the form 5a + 3? No.
Put b = 2. n = 18. Is it of the form 5a + 3? Yes.

When 18 is divided by 5, it gives a remainder of 3. When it is divided by 7, it gives a remainder if 4.

Next such number will be 35 + 18 because 35 will be divisible by 5 as well as 7 and whatever is the remainder from 18, will still be the remainder
Next will be 35*2 + 18
and so on...
Difference between such numbers will be a multiple of 35 so your answer is 35.
Note: Actually, because of this reasoning, you just had to take the LCM. You didn't even need to find the first such number!

I have discussed this topic a little more in detail here: http://gmatclub.com/forum/good-problem-90442-20.html#p814507
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Director
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Location: India
Re: When the positive integer A is divided by 5 and 7, the [#permalink]

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06 Oct 2013, 19:06
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Expert's post
mrinal2100 wrote:
When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

The easiest way to approach these problems is by taking an example

1. The first choice is 18. Take it as B
2. The next choice is 53. Take it as A
3. A-B=35

c is the only choice that is correct.
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Kudos [?]: 557 [2], given: 16

Director
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When the positive integer A is divided by 5 and 7, the [#permalink]

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05 Dec 2017, 16:38
mrinal2100 wrote:
When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

A=5q+3
5q+3=7p+4
5q=7p+1
least value of q=3
least value of p=2
A=18
18+5*7=53=B
18-53=-35
35
C

Kudos [?]: 306 [0], given: 17

When the positive integer A is divided by 5 and 7, the   [#permalink] 05 Dec 2017, 16:38
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