1.59 (1) $f(x)$ is a monovariate quartic polynomial with integer coefficients, i.e.,
$$
f(x)=c_{4} x^{4}+c_{3} x^{3}+c_{2} x^{2}+c_{1} x+c_{0},
$$

where $c_{0}, c_{1}, c_{2}, c_{3}, c_{4}$ are all integers. Prove that if $a, b$ are integers and $a > b$, then $a - b$ divides $f(a) - f(b)$.
(2) Two people, A and B, are doing homework in the classroom.
A asks B: “How many problems have you finished?”
B replies: “The number of problems I have finished is a positive integer and is a root of a monovariate quartic polynomial with integer coefficients. Can you guess it?”
Then, A tries substituting 7 into the polynomial and gets a value of 77. At this point, B says: “The number of problems I have finished is more than 7.”
A says: “Alright, I will try a larger integer.” So A substitutes $B$ into the polynomial and gets a value of 85.
B looks and says: “The number of problems I have finished is more than $B$.” After some thought and calculation, A finally determines the number of problems B has finished.
Try to find the number of problems B has finished based on the conversation.
(China Hebei Province Mathematics Competition, 1979)
[Solution]
$$
\begin{array}{l} 
\text { (1) } f(a)-f(b) \\
=\left(c_{4} a^{4}+c_{3} a^{3}+c_{2} a^{2}+c_{1} a+c_{0}\right)-\left(c_{4} b^{4}+c_{3} b^{3}+c_{2} b^{2}\right. \\
\left.\quad+c_{1} b+c_{0}\right) \\
= c_{4}\left(a^{4}-b^{4}\right)+c_{3}\left(a^{3}-b^{3}\right)+c_{2}\left(a^{2}-b^{2}\right)+c_{1}(a-b)
\end{array}
$$

Since $a-b \mid a^{k}-b^{k}, k \in N$, then
$$
a-b \mid f(a)-f(b) \text {. }
$$
(2) Let the monic quartic polynomial with integer coefficients that Yi mentioned be $f(x)$, and the number of problems Yi did be
A.
According to the problem, we have
$$
f(7)=77, \quad f(B)=85, \quad f(A)=0 .
$$

And
$$
7<B<A \text {. }
$$

Using the result from (1), we have
$$
\begin{array}{l}
A-7 \mid f(A)-f(7)=-77 \\
A-B \mid f(A)-f(B)=-85 \\
B-7 \mid f(B)-f(7)=8
\end{array}
$$

From (1), we know
$$
A-7=1,7,11,77 \text {. }
$$

Thus
$$
\begin{array}{c}
A=8, \quad 14, \quad 18, \quad 84 . \\
B-7=1, \quad 2, \quad 4, \quad 8 . \\
B=8,9,11,15 .
\end{array}
$$

From (3), we know
"Thus
Considering the different combinations of $A$ and $B$, only when $A=14, B=9$, it satisfies
$$
A-B \mid-85 \text {. }
$$

Therefore, $A=14$, meaning Yi did 14 problems.