TITLE: Finding the number of polynomial when $f(a) , f(b)$ is given
QUESTION [7 upvotes]: Find the number of polynomials given by $f(x)$ , with integral coefficients. Such that $$f(7) =11; f(11)=13$$
Please suggest the solution of this..
Thanks ..

REPLY [3 votes]: $f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_0$
$f(7)=a_n7^n+a_{n-1}7^{n-1}+\dots a_0=11$
$f(11)=a_n11^n+a_{n-1}11^{n-1}+\dots a_0=13$
$f(11)-f(7)=a_n(11^n-7^n) +a_{n-1}(11^{n-1}-7^{n-1})+ \dots a_1\cdot4=2$
Now you know that $a-b|a^n-b^n$, from here.
$4|f(11)-f(7) \implies f(11)-f(7)=4k$
But $\dots 4 \not| 2$, conclude.