An integer function maps a real number to an integer value. In this post, we’re going to discuss three integer functions that is widely applied in Number Theory — Floor Function, Ceiling Function and the Sawtooth Function.

**Summary**

For a real number ,

- Floor Function returns the largest integer less than or equal to , denote as .
- Ceiling Function returns the smallest integer larger than or equal to , denote as .
- Sawtooth Function returns the fractional part of , denote as .

**Properties**

Note: The variable in this section is assumed to be an integer.

**Problem Solving**

Solve the equation for non-zero solution.

Since , then . Substitute , we have . Finally,

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