Find the highest power if 5 in 100!.

Options

A. 19
B. 22
C. 25
D. None of these

Answer

To find the highest power of 5 in \(100!\), we can use the formula for finding the highest power of a prime \(p\) in \(n!\):

\[ \text{Highest power of } p \text{ in } n! = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \ldots \]

where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).

### Step-by-step Solution:

1. **Identify \(n\) and \(p\)**:
- Here, \(n = 100\) and \(p = 5\).

2. **Calculate \(\left\lfloor \frac{100}{5} \right\rfloor\)**:
\[ \left\lfloor \frac{100}{5} \right\rfloor = \left\lfloor 20 \right\rfloor = 20 \]

3. **Calculate \(\left\lfloor \frac{100}{5^2} \right\rfloor\)**:
\[ \left\lfloor \frac{100}{25} \right\rfloor = \left\lfloor 4 \right\rfloor = 4 \]

4. **Calculate \(\left\lfloor \frac{100}{5^3} \right\rfloor\)**:
\[ \left\lfloor \frac{100}{125} \right\rfloor = \left\lfloor 0.8 \right\rfloor = 0 \]

5. **Sum the results**:
- Now, we add the results from steps 2, 3, and 4:
\[ 20 + 4 + 0 = 24 \]

### Conclusion:
The highest power of 5 in \(100!\) is \(24\).