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

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

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New post 10 Mar 2016, 03:29
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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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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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

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New post 10 Mar 2016, 05:02
3
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

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

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New post 10 Mar 2016, 08:36
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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..
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Re: When a positive integer A is divided by 5 and 7, the remainders obtain  [#permalink]

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New post 17 Mar 2016, 04:52
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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Re: When a positive integer A is divided by 5 and 7, the remainders obtain  [#permalink]

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New post 18 Oct 2016, 20:16
1
2
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.

Answer: E.

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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Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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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, 20:42
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