Find the number of highest power of 7 in 777!

Options

A. a)128
B. b)130
C. c)126
D. d)125

Answer

To find the highest power of 7 in 777!, we can use the formula for determining the highest power of a prime \( p \) in \( n! \):

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

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, \( n = 777 \) and \( p = 7 \).

2. **Calculate \( \left\lfloor \frac{777}{7} \right\rfloor \)**:
\[ \left\lfloor \frac{777}{7} \right\rfloor = \left\lfloor 111 \right\rfloor = 111 \]

3. **Calculate \( \left\lfloor \frac{777}{7^2} \right\rfloor \)**:
\[ \left\lfloor \frac{777}{49} \right\rfloor = \left\lfloor 15.8571 \right\rfloor = 15 \]

4. **Calculate \( \left\lfloor \frac{777}{7^3} \right\rfloor \)**:
\[ \left\lfloor \frac{777}{343} \right\rfloor = \left\lfloor 2.267 \right\rfloor = 2 \]

5. **Calculate \( \left\lfloor \frac{777}{7^4} \right\rfloor \)**:
\[ \left\lfloor \frac{777}{2401} \right\rfloor = \left\lfloor 0.323 \right\rfloor = 0 \]

6. **Sum the results**:
- Now, we add all the results together:
\[ 111 + 15 + 2 + 0 = 128 \]

### Final Answer:
The highest power of 7 in \( 777! \) is **128**.

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