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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.

\[\lfloor1\rfloor\times\lfloor2\rfloor\times\lfloor3\rfloor\times\lfloor4\rfloor\times\cdots \times\lfloor100\rfloor\]

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**:

\( \lceil \cdot \rceil \) denotes the ceiling function.

\( \{ \cdot \} \) denotes the fractional part function.

**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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