Find $C$ such that $x^2 - 47x - C = 0$ has integer roots, and further conditions?

Question

Have been working on this for years. Need a system which proves that there exists a number $C$ which has certain properties. I will give a specific example, but am looking for a system which could possibly be generalized. Find (or prove there exists) $C$, such that the quadratic $x^2 - 47x - C = 0$ has integer roots, and furthermore, $C$ must have ALL OF $2$, $3$ and $5$ as its only prime factors (though each of these can be to any positive integer power).

Answer 1

The statement is false.

C must be the product for two positive integers whose difference is 47

1 and 48 which has product 48 and factors as (x + 1 )(x - 48 )

2 and 49 which has product 98 (x + 2)(x - 49 )

3 and 50 which has product 150 (x+3) (x-50)

4 and 51 which has product 204 (x+4)(x-51)

5 and 52 which has product 260 (x+5)(x-52)

However, 98 = 2 * 49 = 2 * 7 * 7

204 = 2 * 102 = 2 * 2 * 51 = 2 * 2 * 3 * 17

260 = 2 * 130 = 2 * 2 * 65 = 2*2* 5 * 13