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# Floor and Ceiling Functions

Floor functions map a real number x to the largest integer less than or equal to x. You can probably guess what the ceiling function does.

# Floor and Ceiling Functions: Level 2 Challenges

$\lfloor1\rfloor\times\lfloor2\rfloor\times\lfloor3\rfloor\times\lfloor4\rfloor\times\cdots \times\lfloor100\rfloor$

Find the trailing number of zeros of the product above.



Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

Find the number of trailing zeros in the following expression.

$1!^2 \times 2!^2 \times \cdots \times 30!^2.$

If $$x$$ is an irrational number, what is the value of $$\left \lceil \left \{ x \right \} \right \rceil$$?

Notations:

True or false:

$$\quad$$ For any real number $$x$$, the value of $$\{x\}$$ can never be 1.

 Notation: $$\{ \cdot \}$$ denotes the fractional part function.

Let $$3^a$$ be the highest power of 3 that divides $$1000!$$. What is $$a$$?

Note: $$1000! = 1 \times 2 \times 3 \ldots \times 999 \times 1000$$.

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