Find the highest power of 6 in 60!

Answer

To find the highest power of 6 in \(60!\), we can break down the problem into finding the highest powers of its prime factors, which are 2 and 3. The highest power of 6 in \(60!\) will be determined by the limiting factor between the highest powers of 2 and 3.

### Step-by-Step Solution:

1. **Identify the Prime Factorization of 6**:
\[ 6 = 2 \times 3 \]
We need to find the highest powers of 2 and 3 in \(60!\).

2. **Calculate the Highest Power of 2 in \(60!\)**:
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 \(p = 2\) and \(n = 60\):
\[ \left\lfloor \frac{60}{2^1} \right\rfloor + \left\lfloor \frac{60}{2^2} \right\rfloor + \left\lfloor \frac{60}{2^3} \right\rfloor + \left\lfloor \frac{60}{2^4} \right\rfloor + \left\lfloor \frac{60}{2^5} \right\rfloor + \left\lfloor \frac{60}{2^6} \right\rfloor \]
Calculating each term:
- \( \left\lfloor \frac{60}{2} \right\rfloor = 30 \)
- \( \left\lfloor \frac{60}{4} \right\rfloor = 15 \)
- \( \left\lfloor \frac{60}{8} \right\rfloor = 7 \)
- \( \left\lfloor \frac{60}{16} \right\rfloor = 3 \)
- \( \left\lfloor \frac{60}{32} \right\rfloor = 1 \)
- \( \left\lfloor \frac{60}{64} \right\rfloor = 0 \) (stop here)

Now, summing these values:
\[ 30 + 15 + 7 + 3 + 1 = 56 \]
Thus, the highest power of 2 in \(60!\) is **56**.

3. **Calculate the Highest Power of 3 in \(60!\)**:
Using the same formula for \(p = 3\):
\[ \left\lfloor \frac{60}{3^1} \right\rfloor + \left\lfloor \frac{60}{3^2} \right\rfloor + \left\lfloor \frac{60}{3^3} \right\rfloor \]
Calculating each term:
- \( \left\lfloor \frac{60}{3} \right\rfloor = 20 \)
- \( \left\lfloor \frac{60}{9} \right\rfloor = 6 \)
- \( \left\lfloor \frac{60}{27} \right\rfloor = 2 \)
- \( \left\lfloor \frac{60}{81} \right\rfloor = 0 \) (stop here)

Now, summing these values:
\[ 20 + 6 + 2 = 28 \]
Thus, the highest power of 3 in \(60!\) is **28**.

4. **Determine the Highest Power of 6 in \(60!\)**:
Since \(6 = 2 \times 3\), the highest power of 6 will be the minimum of the highest powers of 2 and 3:
\[ \text{Highest power of } 6 = \min(56, 28) = 28 \]

### Final Answer:
The highest power of 6 in \(60!\) is **28**.