We inspect those polynomials whose coefficients are integers. We say that an integer m is a divisor of a polynomial P if some value of P is divisible by m. Our main result is that the common divisors of any several polynomials are exactly the divisors of a single polynomial. This is extended to prove that the set of prime numbers for which a system of polynomial equations in multiple variables is solvable is exactly the set of prime divisors of some polynomial in one variable. In addition, we prove results on how often a prime number is a common prime divisor of some polynomials – we prove a tight lower bound for this so-called density, and under additional assumptions give a formula for this density. Our work generalizes previous results, and we propose several ideas for further research.