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When the positive integer A is divided by 5 and 7
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Updated on: 06 Jun 2013, 06:22
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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 AB? (A) 12 (B) 24 (C) 35 (D) 16 (E) 30
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Originally posted by BDSunDevil on 06 Jan 2012, 12:09.
Last edited by Bunuel on 06 Jun 2013, 06:22, edited 3 times in total.
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Re: no.prop
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13 Jan 2012, 06:55
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 AB? (A) 12 (B) 24 (C) 35 (D) 16 (E) 30 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): \(B=35n+18\); \(AB=(35m+18)(35n+18)=35(mn)\) > thus AB must be a multiple of 35. Answer: C. More about this concept: manhattanremainderproblem93752.html?hilit=derive#p721341goodproblem90442.html?hilit=derive#p722552Hope it helps.
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Re: When the positive integer A is divided by 5 and 7
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04 Aug 2014, 21:27
Although there are methods.. this particular problem does not require any methods..
A divided by 5,7 gives 3 and 4 B divided by 5,7 gives 3 and 4.
means A and B are equidistant from the multiples of 5,7. the LCM of 5 and 7 is 35, so numbers like A & B repeat after every 35 numbers. so 35 is the answer.



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Re: When the positive integer A is divided by 5 and 7
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06 Aug 2014, 03:05
With the remainders remaining same for both the required integers, it means there difference should be divisible by both 5 & 7 Out of all the five given options, only 35 stands out Answer = C
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Re: When the positive integer A is divided by 5 and 7
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14 Dec 2015, 00:40
Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer. 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 AB? (A) 12 (B) 24 (C) 35 (D) 16 (E) 30 By the conditions we may put A=5K+3 = 7L+4, B=5M+3 = 7N+4. So we have AB= 5K+35M3 = 5*(KM)= 7L+4  7N4=7*(LN). We may conclude that AB is divisible by 5 and 7. That means 35 is a factor of AB. So the answer is (C).
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When the positive integer A is divided by 5 and 7
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03 Jun 2017, 21:39
BDSunDevil 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 AB?
(A) 12 (B) 24 (C) 35 (D) 16 (E) 30 because the difference between any values of A and B will always be a multiple of the product of their common divisors, 5 and 7, the correct answer is 35 C



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Re: When the positive integer A is divided by 5 and 7
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16 Jun 2018, 09:59
BDSunDevil 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 AB?
(A) 12 (B) 24 (C) 35 (D) 16 (E) 30 Since when A is divided by 5, the remainder is 3, A can be values such as: 3, 8, 13, 18, ... Since when A is divided by 7, the remainder is 4, A can be values such as: 4, 11, 18, ... We see that the smallest number A can be is 18; the next value for A is 18 plus the LCM of 5 and 7, that is, A = 18 + 35 = 53. And we can keep adding 35 to obtain succeeding values of A. That is, A can be values such as: 18, 53, 88, 123, … Since B has the same condition as A, then B can be any of the values above. Therefore, the difference between A and B must be a multiple of 35 (for example, if A = 88 and B = 18, A  B = 70 = 2 x 35). Alternate Solution: Since both A and B produce the same remainder when divided by 5, A  B must be divisible by 5. (The reason is the following: Since A produces a remainder of 3 when divided by 5, A must be expressible as 5p + 3 for some integer p. Since B produces a remainder of 3 when divided by 5, B must be expressible as 5q + 3 for some integer q. Then, A  B can be expressed as 5p + 3  (5q + 3) = 5p  5q = 5(p  q); which is a multiple of 5) Similarly, since both A and B produce the same remainder when divided by 7, A  B must be divisible by 7. Since A  B is divisible by both 5 and 7, it must be divisible by LCM of 5 and 7, which is 35. Answer: C
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Re: When the positive integer A is divided by 5 and 7
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18 Jun 2018, 08:32
sshrivats wrote: Although there are methods.. this particular problem does not require any methods..
A divided by 5,7 gives 3 and 4 B divided by 5,7 gives 3 and 4.
means A and B are equidistant from the multiples of 5,7. the LCM of 5 and 7 is 35, so numbers like A & B repeat after every 35 numbers. so 35 is the answer. I really like this method mentioned above, but myself used another approach, more time consuming. From given information, we know that A must be 18 (at least) to satisfy both conditions. Likewise, we know B must be 53( at least) to satisfy both conditions. So, we have 5318=35 (C)
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Re: When the positive integer A is divided by 5 and 7
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18 Jul 2019, 23:40
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