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Let $F(x)$ be a polynomial function of degree $2018$ and leading coefficient unity such that $F(0) = 2017$, $F(1) = 2016$, $F(2) = 2015, \ldots$, $F(2017) = 0$. The value of $F(2018)$ is of the form $n! - a$. where $n, a \in \mathbb{N}$ and $a$ is least possible value, then $n + a$ is equal to what?

I am not getting any start .

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    $n+a$ is equal to what? – 2017-02-18
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    Possible duplicate : http://math.stackexchange.com/questions/2011961/if-p-is-a-monic-polynomial-of-degree-2015-such-that-pi-2014-i-quad-forall/2011971#2011971 – 2017-02-18

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Consider the polynomial function $g(x)=x+f(x)-2017$.

The degree of $g$ is $2018$. Based on the information given, $0,1,2, \ldots , 2017$ are all roots of $g$.

Thus $$g(x)=x(x-1)(x-2) \dotsb (x-2017)$$

Can you take it from here?