Find the largest power of 7 that can exactly divide 777!.

Answer

To find the largest power of 7 that can exactly divide 777!, we will use the formula for finding the highest power of a prime \( p \) that divides \( n! \):

\[ \text{Highest power of } p \text{ in } n! = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]

In this case, \( n = 777 \) and \( p = 7 \).

### Step-by-Step Solution:

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

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

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

4. **Calculate \( \left\lfloor \frac{777}{7^4} \right\rfloor \)**:
\[ 7^4 = 2401 \quad \Rightarrow \quad \frac{777}{2401} \approx 0.323 \]
Since \( 0 < 1 \), \( \left\lfloor \frac{777}{2401} \right\rfloor = 0 \).

5. **Sum the results**:
Now we sum all the values obtained:
\[ 111 + 15 + 2 + 0 = 128 \]

Thus, the largest power of 7 that can exactly divide \( 777! \) is **128**.

### Final Answer:
The largest power of 7 that can exactly divide \( 777! \) is **128**.