Let \(P(x)\) be a polynomial with integer coefficients that satisfies \(P(17)=10\) and \(P(24)=17.\) Given that \(P(n)=n+3\) has two distinct integer solutions \(n_1\) and \(n_2,\) find the product \(n_1\cdot n_2.\)

(第二十三届AIME2 2005 第13题)