Find the highest power of 30 in 50!

Answer

To find the highest power of 30 in 50!, we first need to factor 30 into its prime factors.

### Step-by-Step Solution:

1. **Factor 30 into Prime Factors**:
\[ 30 = 2 \times 3 \times 5 \]
The prime factors of 30 are 2, 3, and 5.

2. **Identify the Limiting Factor**:
In this case, the limiting factor will be the highest power of 5 in 50!, since 5 is the largest prime factor in the factorization of 30.

3. **Calculate the Highest Power of 5 in 50!**:
To find the highest power of a prime \( p \) in \( n! \), we use the formula:
\[ \text{Highest power of } p \text{ in } n! = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]
For \( p = 5 \) and \( n = 50 \):
- Calculate \( \left\lfloor \frac{50}{5} \right\rfloor \):
\[ \left\lfloor \frac{50}{5} \right\rfloor = 10 \]
- Calculate \( \left\lfloor \frac{50}{25} \right\rfloor \):
\[ \left\lfloor \frac{50}{25} \right\rfloor = 2 \]
- Higher powers of 5 (like \( 5^3 = 125 \)) do not contribute since \( 50 < 125 \).

4. **Sum the Contributions**:
Now, we add the contributions from the powers of 5:
\[ 10 + 2 = 12 \]

5. **Conclusion**:
The highest power of 30 in 50! is determined by the limiting factor, which is the highest power of 5. Therefore, the highest power of 30 in 50! is:
\[ \text{Highest power of 30 in } 50! = 12 \]