A problem based on number theory(congruence relation)

Question

Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying $f(x)=1 \pmod{x-1}$, $f(x)=0 \pmod{x-3}$.

Which of the following statements are true? 1. $S$ is empty 2. $S$ is a Singleton 3. $S$ is a finite non empty set 4. $S$ is countably infinite

If we put x=7 then gcd(x-1,x-3) does not divide 1. So according to me option 1 is correct. But I have no general method to prove it. Please help me.

Answer 1

Hint: As one can always divide by a monic polynomial in $\mathbf Z[x]$, the second hypothesis means $f(x)$ is a multiple of $x-3$ : $$f(x)=(x-3)g(x),\quad g(x)\in\mathbf Z[x].$$ What can you deduce for $f(1)$?

Answer 2

When $x$ is odd, then $f(x)$ has to be both odd and even which is impossible, so $S$ is empty.