Question

Let P(x) be a non-zero polynomial with integer coefficients. If P(n) is divisible by n for each positive integer n, what is the value of P(0)?
Answer is 00.
Provide solution.
10 points.

Answer 1

Let p(x)=a0+a1x+a2x2+...+akxk

Note that p(0)=a0

p(n)=a0+a1n+a2n2+...+aknk

Evidently, p(n)≡a0mod n for all nn

But, p(n)≡0mod n for all n

⟹a0=0

⟹p(0)=0

However, if the condition was p(n)≡0mod n for all n≤N where N is some positive integer, there are 2 solutions: a0= or lcm(1,2,3,...,N)