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If p and q are positive integers such that when they are divided by 5, [#permalink]
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Updated on: 04 Jul 2017, 06:02
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If p and q are positive integers such that when they are divided by 5, the remainder is 3 for each; and when they are divided by 9, the remainder is 4 for each. If q>p, then which of the following must be a factor of q  p ? A) 12 B) 20 C) 27 D) 36 E) 45
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Originally posted by franz711 on 04 Jul 2017, 05:22.
Last edited by Bunuel on 04 Jul 2017, 06:02, edited 1 time in total.
Added the OA.



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Re: If p and q are positive integers such that when they are divided by 5, [#permalink]
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04 Jul 2017, 05:43
franz711 wrote: If p and q are positive integers such that when they are divided by 5, the remainder is 3 for each; and when they are divided by 9, the remainder is 4 for each. If q>p, then which of the following must be a factor of q  p ?
A) 12 B) 20 C) 27 D) 36 E) 45 When p is divided by 5, the remainder is 3: p= 5m + 3, so it can be 3, 8, 13, 18, ... When p is divided by 9, the remainder is 4: p= 9n + 4, so it can be 4, 13, 22, 31, ... There is a way to derive general formula for p (of a type p = kx + r, where x is divisor and r is a remainder) based on above two statements:Divisor x would be the least common multiple of above two divisors 5 and 9, hence x=45. Remainder r would be the first common integer in above two patterns, hence r = 13. Therefore general formula based on both statements is p = 45x + 13. (check HERE to know to to derive general formula from these two) Similarly, general formula for q will be 45y + 13. q  p = (45y + 13)  (45x + 13) = 45(y  x). Answer: E.
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Re: If p and q are positive integers such that when they are divided by 5, [#permalink]
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04 Jul 2017, 05:47
franz711 wrote: If p and q are positive integers such that when they are divided by 5, the remainder is 3 for each; and when they are divided by 9, the remainder is 4 for each. If q>p, then which of the following must be a factor of q  p ?
A) 12 B) 20 C) 27 D) 36 E) 45 I went the long way , but it wasn't too time consuming because these numbers turn out to be very manageable: 1. When positive integers p and q are divided by 5, the remainder for each is 3. p = 5a + 3 q = 5b + 3 2. When positive integers p and q are divided by 9, the remainder for each is 4 p = 9c + 4 q = 9d + 4 3. Possible values for p and q for both sets of equations, and from each list of possibilities we need two values that match because q > p: #1: 3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58#2: 4, 13, 22, 31, 40, 49, 58q = 58, p = 13 q  p = (58  13) = 45 Answer E
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If p and q are positive integers such that when they are divided by 5, [#permalink]
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Updated on: 08 Jul 2018, 17:30
franz711 wrote: If p and q are positive integers such that when they are divided by 5, the remainder is 3 for each; and when they are divided by 9, the remainder is 4 for each. If q>p, then which of the following must be a factor of q  p ?
A) 12 B) 20 C) 27 D) 36 E) 45 because divisor ratio of 5:9≈1:2, assume quotient ratio inversely=2:1 [(p3)/5)]/[(p4)/9]=2 p=13 13+(5*9)=58=q qp=5813=45 E
Originally posted by gracie on 04 Jul 2017, 13:41.
Last edited by gracie on 08 Jul 2018, 17:30, edited 1 time in total.



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Re: If p and q are positive integers such that when they are divided by 5, [#permalink]
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22 Aug 2017, 06:42
I used this rule: remainder can be added or subtracted directly when adding or subtracting two dividends (the excess needs to be corrected after)
So p and q has the same remainder when divided by 5, therefore qp will have the remainder of 0, which means qp is divisible by 5. The same goes for 9.
E) 45 is the only answer that is a multiple of 9 and 5, so E is the right answer.



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Re: If p and q are positive integers such that when they are divided by 5, [#permalink]
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08 Jul 2018, 12:34
I went this way: p=5x+3 p=9x'+4 q=5y+3 q=9y'+4 => qp =5(yx) = 9(y'x') =>9*5=45 is a factor of qp same as NamVu1990 Hope it will help someone
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Re: If p and q are positive integers such that when they are divided by 5,
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