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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