Suppose that f(x) = x

3 px2 + qx and g(x) = 3x

2 2px + q for some positive

integers p and q.

(a) If p = 33 and q = 216, show that the equation f(x) = 0 has three distinct

integer solutions and the equation g(x) = 0 has two distinct integer solutions.

(b) Suppose that the equation f(x) = 0 has three distinct integer solutions and the

equation g(x) = 0 has two distinct integer solutions. Prove that

(i) p must be a multiple of 3,

(ii) q must be a multiple of 9,

(iii) p

2 3q must be a positive perfect square, and

(iv) p

2 4q must be a positive perfect square.

(c) Prove that there are infinitely many pairs of positive integers (p, q) for which

the following three statements are all true:

The equation f(x) = 0 has three distinct integer solutions.

The equation g(x) = 0 has two distinct integer solutions.

The greatest common divisor of p and q is 3 (that is, gcd(p, q) = 3).