Example 11 (2003 Tianjin Competition Question) Let the quadratic function $f(x)=a x^{2}+b x+c$ (where $a, b, c$ are integers), and 4 students calculate the function values. Student A gets: $f(7)=-1$; Student B gets: $f(1)=3$; Student C gets: $f(4)=-4$; Student D gets: $f(2)=4$. Among them, only 1 student made a calculation error, then the student who made the error is ( ).
A. A
B. B
C. C
D. D
Solution: Choose B. Reason: Since $f(m)-f(n)=(m-n)(a m+a n+b)$, then $(m-n) \mid (f(m)-f(n))$. $(4-2) \mid (-4-4)$.
Therefore, Yi calculated incorrectly.