If `300!=3^(m)xx`an integer, then find the value of m .

Options

A. m=148
B. m=150
C. It is equivalent to number of n is 150!`=2^(n-2)xx`an integer
D. `m=.^(150)C_(2)`

Explanation

To find the value of \( m \) such that \( 300! = 3^m \times k \) (where \( k \) is an integer), we need to determine the highest power of 3 that divides \( 300! \). This can be calculated using the formula for finding the highest power of a prime \( p \) that divides \( n! \):

\[ m = \sum_{i=1}^{\infty} \left\lfloor \frac{n}{p^i} \right\rfloor \]

In our case, \( n = 300 \) and \( p = 3 \).

### Step 1: Calculate \( \left\lfloor \frac{300}{3} \right\rfloor \)

\[ \left\lfloor \frac{300}{3} \right\rfloor = \left\lfloor 100 \right\rfloor = 100 \]

### Step 2: Calculate \( \left\lfloor \frac{300}{3^2} \right\rfloor \)

\[ \left\lfloor \frac{300}{9} \right\rfloor = \left\lfloor 33.33 \right\rfloor = 33 \]

### Step 3: Calculate \( \left\lfloor \frac{300}{3^3} \right\rfloor \)

\[ \left\lfloor \frac{300}{27} \right\rfloor = \left\lfloor 11.11 \right\rfloor = 11 \]

### Step 4: Calculate \( \left\lfloor \frac{300}{3^4} \right\rfloor \)

\[ \left\lfloor \frac{300}{81} \right\rfloor = \left\lfloor 3.70 \right\rfloor = 3 \]

### Step 5: Calculate \( \left\lfloor \frac{300}{3^5} \right\rfloor \)

\[ \left\lfloor \frac{300}{243} \right\rfloor = \left\lfloor 1.23 \right\rfloor = 1 \]

### Step 6: Calculate \( \left\lfloor \frac{300}{3^6} \right\rfloor \)

\[ \left\lfloor \frac{300}{729} \right\rfloor = \left\lfloor 0.41 \right\rfloor = 0 \]

Since \( \left\lfloor \frac{300}{3^6} \right\rfloor = 0 \), we stop here.

### Step 7: Sum all the values calculated

Now we sum all the values obtained:

\[ m = 100 + 33 + 11 + 3 + 1 = 148 \]

### Conclusion

Thus, the value of \( m \) is \( 148 \).

### Final Answer

Hence, \( m = 148 \).

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