the exponent of `15` in `100!,` is

Options

A. 12
B. 24
C. 36
D. 48

Answer

To find the exponent of 15 in \(100!\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Factorization of 15**:
- The number 15 can be factored into its prime factors: \(15 = 5 \times 3\).
- To find the exponent of 15 in \(100!\), we need to determine how many times both 5 and 3 appear in the factorization of \(100!\).

2. **Finding the Exponent of 5 in \(100!\)**:
- We use the formula to find the exponent of a prime \(p\) in \(n!\):
\[ \text{Exponent of } p = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \ldots \]
- For \(p = 5\) and \(n = 100\):
\[ \text{Exponent of } 5 = \left\lfloor \frac{100}{5} \right\rfloor + \left\lfloor \frac{100}{25} \right\rfloor + \left\lfloor \frac{100}{125} \right\rfloor \]
- Calculating each term:
- \(\left\lfloor \frac{100}{5} \right\rfloor = 20\)
- \(\left\lfloor \frac{100}{25} \right\rfloor = 4\)
- \(\left\lfloor \frac{100}{125} \right\rfloor = 0\) (since \(125 > 100\))
- Therefore, the total exponent of 5 in \(100!\) is:
\[ 20 + 4 + 0 = 24 \]

3. **Finding the Exponent of 3 in \(100!\)**:
- Now, we apply the same formula for \(p = 3\):
\[ \text{Exponent of } 3 = \left\lfloor \frac{100}{3} \right\rfloor + \left\lfloor \frac{100}{9} \right\rfloor + \left\lfloor \frac{100}{27} \right\rfloor + \left\lfloor \frac{100}{81} \right\rfloor \]
- Calculating each term:
- \(\left\lfloor \frac{100}{3} \right\rfloor = 33\)
- \(\left\lfloor \frac{100}{9} \right\rfloor = 11\)
- \(\left\lfloor \frac{100}{27} \right\rfloor = 3\)
- \(\left\lfloor \frac{100}{81} \right\rfloor = 1\)
- Therefore, the total exponent of 3 in \(100!\) is:
\[ 33 + 11 + 3 + 1 = 48 \]

4. **Finding the Exponent of 15**:
- The exponent of 15 in \(100!\) is determined by the limiting factor, which is the minimum of the exponents of 5 and 3:
\[ \text{Exponent of } 15 = \min(\text{Exponent of } 5, \text{Exponent of } 3) = \min(24, 48) = 24 \]

### Final Answer:
The exponent of 15 in \(100!\) is **24**.

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