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10 Mar 2016, 04:29
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E
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Difficulty:
35%
(medium)
Question Stats:
75% (02:03) correct
25%
(02:21)
wrong
based on 305
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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
(A) 12
(B) 16
(C) 24
(D) 30
(E) 35
10 Mar 2016, 06:02
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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
Answer E
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
18 Oct 2016, 21:16
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Bunuel
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.
