What is the maximum value of m such that `7^m` divides into 14! Evenly? (n! means `1 cdot 2 cdot 3 cdot …..cdot n`)

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the maximum value of \( m \) such that \( 7^m \) divides \( 14! \) evenly, we can use the formula for finding the highest power of a prime \( p \) that divides \( n! \):

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

In this case, \( n = 14 \) and \( p = 7 \).

### Step 1: Calculate \( \left\lfloor \frac{14}{7^1} \right\rfloor \)

\[ \left\lfloor \frac{14}{7^1} \right\rfloor = \left\lfloor \frac{14}{7} \right\rfloor = \left\lfloor 2 \right\rfloor = 2 \]

### Step 2: Calculate \( \left\lfloor \frac{14}{7^2} \right\rfloor \)

\[ \left\lfloor \frac{14}{7^2} \right\rfloor = \left\lfloor \frac{14}{49} \right\rfloor = \left\lfloor 0.2857 \right\rfloor = 0 \]

### Step 3: Sum the results

Now, we sum the results from Step 1 and Step 2:

\[ \text{Total} = 2 + 0 = 2 \]

Thus, the maximum value of \( m \) such that \( 7^m \) divides \( 14! \) evenly is \( m = 2 \).

### Final Answer

The maximum value of \( m \) such that \( 7^m \) divides \( 14! \) evenly is \( \boxed{2} \).

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