What will be the maximum power of 8 that exactly divides 25! ?

Options

A. 6
B. 7
C. 8
D. 9

Answer

To find the maximum power of 8 that exactly divides 25!, we can follow these steps:

### Step 1: Understand the relationship between 8 and 2
Since \( 8 = 2^3 \), we need to find the maximum power of 2 in \( 25! \) first.

### Step 2: Calculate the power of 2 in \( 25! \)
To find the power of a prime \( p \) in \( n! \), we can use the formula:
\[ \text{Power of } p \text{ in } n! = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \ldots \]
For our case, \( n = 25 \) and \( p = 2 \).

### Step 3: Apply the formula for \( p = 2 \)
Now we will calculate the contributions from each term:
1. \( \left\lfloor \frac{25}{2} \right\rfloor = 12 \)
2. \( \left\lfloor \frac{25}{4} \right\rfloor = 6 \)
3. \( \left\lfloor \frac{25}{8} \right\rfloor = 3 \)
4. \( \left\lfloor \frac{25}{16} \right\rfloor = 1 \)

### Step 4: Sum the contributions
Now we sum these values:
\[ 12 + 6 + 3 + 1 = 22 \]
So, the power of 2 in \( 25! \) is 22.

### Step 5: Convert power of 2 to power of 8
Since \( 8 = 2^3 \), we need to find how many times \( 8 \) can divide \( 25! \):
\[ \text{Power of } 8 = \left\lfloor \frac{\text{Power of } 2}{3} \right\rfloor = \left\lfloor \frac{22}{3} \right\rfloor = 7 \]

### Conclusion
Thus, the maximum power of 8 that exactly divides \( 25! \) is \( 7 \).

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