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

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

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New post 08 Nov 2010, 09:59
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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
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Re: number prop  [#permalink]

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New post 08 Nov 2010, 10:17
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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.

Answer: C.

More about this concept:
manhattan-remainder-problem-93752.html?hilit=derive#p721341
good-problem-90442.html?hilit=derive#p722552

Hope it helps.
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Re: number prop  [#permalink]

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New post 08 Nov 2010, 11:07
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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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Re: When the positive integer A is divided by 5 and 7, the  [#permalink]

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New post 06 Oct 2013, 20:06
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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


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

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New post Updated on: 26 Jan 2018, 11:44
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


if A and B, when divided by the same divisors, leave the same remainders,
then the difference between them will always be a multiple of the product of those divisors
5*7=35
C

Originally posted by gracie on 05 Dec 2017, 17:38.
Last edited by gracie on 26 Jan 2018, 11:44, edited 1 time in total.
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Re: When the positive integer A is divided by 5 and 7, the  [#permalink]

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New post 26 Jan 2018, 09:45
SravnaTestPrep wrote:
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.


I chose the same approach but only a further tip to speed up:
When you found the value of A you can simply try to add every value from the answer choices: if the number obtained satisfy the initial divisibility conditions then you have found the correct answer :grin:
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Re: When the positive integer A is divided by 5 and 7, the  [#permalink]

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New post 01 Oct 2018, 01:07
VeritasKarishma

Mam, I did it this way, is this a correct way?

A=5Q+3 & A=7s+4
so equating we get

5Q=7s+1

Which is for both A&B so the numbers derived are 18 and 53 solving for the above, hence the factor of 53-18= 35

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

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New post 02 Oct 2018, 04:55
BigUD94 wrote:
VeritasKarishma

Mam, I did it this way, is this a correct way?

A=5Q+3 & A=7s+4
so equating we get

5Q=7s+1

Which is for both A&B so the numbers derived are 18 and 53 solving for the above, hence the factor of 53-18= 35

C


It seems that basically you have found 2 numbers which satisfy both conditions and taken their difference (which would need to be a multiple of at least one of the choices). It is correct. I hope you understand why it is correct. It you are not sure, check out the link I have given above in my post.
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Re: When the positive integer A is divided by 5 and 7, the   [#permalink] 02 Oct 2018, 04:55
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