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If p and q are positive integers such that when they are divided by 5,

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If p and q are positive integers such that when they are divided by 5,  [#permalink]

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New post Updated on: 04 Jul 2017, 05:02
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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

Originally posted by franz711 on 04 Jul 2017, 04:22.
Last edited by Bunuel on 04 Jul 2017, 05:02, 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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New post 04 Jul 2017, 04:43
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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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New post 04 Jul 2017, 04:47
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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, 58

q = 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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New post Updated on: 08 Jul 2018, 16:30
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
[(p-3)/5)]/[(p-4)/9]=2
p=13
13+(5*9)=58=q
q-p=58-13=45
E

Originally posted by gracie on 04 Jul 2017, 12:41.
Last edited by gracie on 08 Jul 2018, 16:30, 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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New post 22 Aug 2017, 05:42
2
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 q-p will have the remainder of 0, which means q-p 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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New post 08 Jul 2018, 11:34
I went this way:
p=5x+3 p=9x'+4
q=5y+3 q=9y'+4

=> q-p =5(y-x) = 9(y'-x') =>9*5=45 is a factor of q-p

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, &nbs [#permalink] 08 Jul 2018, 11:34
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