The exponent of 3 in 331! is :

Options

A. a. 15
B. b. 2
C. c. 11
D. d. none of these

Answer

To find the exponent of 3 in \( 331! \), we can use the formula for finding the exponent of a prime \( p \) in \( n! \):

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

In this case, \( n = 331 \) and \( p = 3 \).

### Step 1: Calculate \( \left\lfloor \frac{331}{3} \right\rfloor \)

\[ \frac{331}{3} = 110.3333 \quad \Rightarrow \quad \left\lfloor 110.3333 \right\rfloor = 110 \]

### Step 2: Calculate \( \left\lfloor \frac{331}{3^2} \right\rfloor \)

\[ \frac{331}{9} = 36.7777 \quad \Rightarrow \quad \left\lfloor 36.7777 \right\rfloor = 36 \]

### Step 3: Calculate \( \left\lfloor \frac{331}{3^3} \right\rfloor \)

\[ \frac{331}{27} = 12.2592 \quad \Rightarrow \quad \left\lfloor 12.2592 \right\rfloor = 12 \]

### Step 4: Calculate \( \left\lfloor \frac{331}{3^4} \right\rfloor \)

\[ \frac{331}{81} = 4.0864 \quad \Rightarrow \quad \left\lfloor 4.0864 \right\rfloor = 4 \]

### Step 5: Calculate \( \left\lfloor \frac{331}{3^5} \right\rfloor \)

\[ \frac{331}{243} = 1.3605 \quad \Rightarrow \quad \left\lfloor 1.3605 \right\rfloor = 1 \]

### Step 6: Calculate \( \left\lfloor \frac{331}{3^6} \right\rfloor \)

\[ \frac{331}{729} = 0.4535 \quad \Rightarrow \quad \left\lfloor 0.4535 \right\rfloor = 0 \]

Since \( \left\lfloor \frac{331}{3^6} \right\rfloor = 0 \), we stop here.

### Step 7: Sum all the values

Now, we sum all the values obtained:

\[ 110 + 36 + 12 + 4 + 1 = 163 \]

Thus, the exponent of 3 in \( 331! \) is \( \boxed{163} \).