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Question

The number of pairs of positive integers (p,q) such that GCD(p,q)=1 and pq+14q9p is an integer are
(correct answer + 2, wrong answer - 0.50)

A
4
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B
9
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C
12
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D
infinite
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Solution

The correct option is A 4
Let x=pq. The problem becomes equivalent to finding all the positive rational numbers x such that x+149x=n for some integer n

This equation can be rewritten into the quadratic equation
9x29xn+14=0, whose discriminant must be a square number in order for the root x to be a rational number.
i.e. =(9n)24×9×14=k2
9n24×14=m2 (Let m2=k29)
9n2m2=23×7
(3nm)(3n+m)=23×7

We know that 3nm and 3n+m must both either be even or odd and since, their product is even, both should be even.
3nm3n+m222×7222×7

This gives us two pairs of n and m: (5,13) and (3,5).
Plugging them into the original quadratic equation,
9x29xn+14=0 and solving for x, gives us
9x29xn+14=0
For n=5
9x245x+14=0
x=143 or x=13

Now, for n=3
Original quadratic equation becomes
9x227x+14=0
x=73 or x=23
Therefore, there are four pairs (a,b) that satisfy the given conditions, namely (1,3),(2,3),(7,3) and (14,3).

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