3. The function $f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x], x \in[0,100]$, then the number of elements in the range of $f(x)$ is ( )
A. 734
B. 733
C. 701
D. 700
3. For $0 \leqslant x \leqslant 100$, let $x=3 n+r$, where $n$ is a non-negative integer, and $r$ is a real number such that $0 \leqslant r<3$. Then $f(x)=[3 n+r]+[6 n+2 r]+\left[5 n+\frac{5 r}{3}\right]+[9 n+3 r]+[12 n+4 r]=35 n+f(r)$.
Therefore, for each $n \geqslant 0$, the number of different integer values that the function $f$ gives in the interval $[n, n+3)$ is the same.
Since $f(x)$ is a step function, and its value jumps at least at one of the points $[x],[2 x],\left[\frac{5 x}{3}\right],[3 x],[4 x]$. The function $f(x)$ has the following 22 jumps in the interval $[0,3)$: $0, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, 1, \frac{6}{5}, \frac{5}{4}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \frac{7}{4}$, $\frac{9}{5}, 2, \frac{9}{4}, \frac{7}{3}, \frac{12}{5}, \frac{5}{2}, \frac{8}{3}, \frac{11}{4}$. Thus, the number of different values that $f(x)$ takes in the interval $[0,3)$ is 22, and the number of different values that $f(x)$ takes in the interval $[0,99)$ is $33 \times 22=726$. Since the number of different values that $f(x)$ takes in the interval $[99,100]$ is the same as the number of different values that $f(x)$ takes in the interval $[0,1]$, and $f(x)$ has 8 jumps in the interval $[0,1]$, the number of different values that $f(x)$ takes in the interval $[99,100]$ is 8.
In summary, the number of different integer values that $f$ takes in the interval $[0,100]$ is $726+8=734$.