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Math Expert V
Joined: 02 Sep 2009
Posts: 58335
When a positive integer A is divided by 5 and 7, the remainders obtain  [#permalink]

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Difficulty:   15% (low)

Question Stats: 76% (01:48) correct 24% (02:13) wrong based on 88 sessions

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When a positive integer A is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. Which of the following is a factor of (A - B)?

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

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

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When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively:
A=5q+3
A can take values 3 , 8 , 13 , 18 ...
and A=7p+4
A can take values 4 , 11 , 18 , 25 ...

Divisor will be LCM of above two divisors 5 and 7 , that is 5*7= 35

Remainder will be the first common integer in above two sequences, therefore 18 .
To fulfill both the given conditions A must be of a type A=35m+18
18, 53, 88, ....

Similarly for positive integer B
B= 35n + 18

A−B =(35m+18)−(35n+18)= 35(m-n)
=> 35 must be a factor of A-B

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Manager  Joined: 25 Dec 2012
Posts: 116
Re: When a positive integer A is divided by 5 and 7, the remainders obtain  [#permalink]

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Bunuel wrote:
When a positive integer A is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. Which of the following is a factor of (A - B)?

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

Since the number divided by 5 is giving remainder as 3 --> The number should end with either 3 or 8.

For the number ending with 3 or 8 we should get a remainder of 4 when divided by 7. Means multiples of 7 should end with either 4 or 9.

Either it can be 14 or 49. To find the next number ending with 4 or 9 for the multiples of 7 is simply add 70 to it. 14 + 70 = 84 or 49 + 70 = 119

Here we got the answer since we are adding 70, the difference is 70. So the answer is E.

What ever illustrated above is pretty simple way but need a good understanding. If so this can be solved in a minute..
Current Student D
Joined: 12 Aug 2015
Posts: 2574
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: When a positive integer A is divided by 5 and 7, the remainders obtain  [#permalink]

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Superb Question ....
here is my approach
First we find the first value of A =>18 the next values will be => 18+p*35 as 35 is the LCM of A and B
now for B too value will be => 18+n*35
hence 35 must be the factor unless A and B are equal.
hence E
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Math Expert V
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Posts: 58335
Re: When a positive integer A is divided by 5 and 7, the remainders obtain  [#permalink]

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3
Bunuel wrote:
When a positive integer A is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. Which of the following is a factor of (A - B)?

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

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

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_________________ Re: When a positive integer A is divided by 5 and 7, the remainders obtain   [#permalink] 05 Feb 2019, 21:42
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