Problem
ALG-B1-M04-P005 Difference of Values
#5
★★☆☆☆ Level 2 of 5
Let \(P(x)\) be a polynomial with integer coefficients. Prove that \(P(17)-P(5)\) is divisible by \(12\).
For every \(k\), \(17^k-5^k\) is divisible by \(17-5\).
For a monomial \(cx^k\), the difference of values is \(c(17^k-5^k)\), which is divisible by \(12\). The sum of such differences is also divisible by \(12\).
This is the lemma for integer-coefficient problems.