7. Let $f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}$, where $a_{i}$ are nonnegative integers for $i=0,1,2, \ldots, n$. If $f(1)=21$ and $f(25)=78357$, find the value of $f(10)$.
7. Ans: 5097
We use the fact that every positive integer can be expressed uniquely in any base $\geq 2$. As $f(1)=21$, we see that the coefficients of $f(x)$ are nonnegative integers less than 25. Therefore $f(25)=78357$, when written in base 25 , allows us to determine the coefficients. Since
$$
78357=7+9 \times 25+5 \times 25^{3},
$$
we obtain $f(x)=7+9 x+5 x^{3}$. Hence $f(10)=5097$.