Problem 2

Find the total number of different integer values taken by

f(x)=\lfloor x\rfloor+\lfloor2x\rfloor+\left\lfloor\frac{5x}{3}\right\rfloor+\lfloor3x\rfloor+\lfloor4x\rfloor

for real

x

with

0\le x\le100

Step 2 of 5: List all possible jump points in one period

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x\in[0,3):\quad x=0,1,2;\quad x=\frac n2;\quad x=\frac{3n}{5};\quad x=\frac n3;\quad x=\frac n4

Detailed analysis

A floor term changes only when its argument is an integer. In $[0,3)$ the candidates are $x=0,1,2$ ; $x=n/2$ for $0\le n\le5$ ; $x=3n/5$ for $0\le n\le4$ ; $x=n/3$ for $0\le n\le8$ ; and $x=n/4$ for $0\le n\le11$ .