Find the highest power of 81 that can divide 1800!.

Answer

To find the highest power of 81 that can divide 1800!, we can follow these steps:

### Step 1: Factor 81
First, we need to express 81 in terms of its prime factors.
\[ 81 = 3^4 \]

### Step 2: Find the highest power of 3 in 1800!
Next, we need to find the highest power of 3 that divides 1800!. This can be done using the formula for finding the highest power of a prime \( p \) in \( n! \):
\[ \text{Highest power of } p \text{ in } n! = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]
In our case, \( n = 1800 \) and \( p = 3 \).

### Step 3: Calculate the contributions of powers of 3
We will calculate the contributions from different powers of 3:

1. For \( k = 1 \):
\[ \left\lfloor \frac{1800}{3^1} \right\rfloor = \left\lfloor \frac{1800}{3} \right\rfloor = \left\lfloor 600 \right\rfloor = 600 \]

2. For \( k = 2 \):
\[ \left\lfloor \frac{1800}{3^2} \right\rfloor = \left\lfloor \frac{1800}{9} \right\rfloor = \left\lfloor 200 \right\rfloor = 200 \]

3. For \( k = 3 \):
\[ \left\lfloor \frac{1800}{3^3} \right\rfloor = \left\lfloor \frac{1800}{27} \right\rfloor = \left\lfloor 66.67 \right\rfloor = 66 \]

4. For \( k = 4 \):
\[ \left\lfloor \frac{1800}{3^4} \right\rfloor = \left\lfloor \frac{1800}{81} \right\rfloor = \left\lfloor 22.22 \right\rfloor = 22 \]

5. For \( k = 5 \):
\[ \left\lfloor \frac{1800}{3^5} \right\rfloor = \left\lfloor \frac{1800}{243} \right\rfloor = \left\lfloor 7.41 \right\rfloor = 7 \]

6. For \( k = 6 \):
\[ \left\lfloor \frac{1800}{3^6} \right\rfloor = \left\lfloor \frac{1800}{729} \right\rfloor = \left\lfloor 2.47 \right\rfloor = 2 \]

7. For \( k = 7 \):
\[ \left\lfloor \frac{1800}{3^7} \right\rfloor = \left\lfloor \frac{1800}{2187} \right\rfloor = \left\lfloor 0.82 \right\rfloor = 0 \]
(We stop here as the value is now 0.)

### Step 4: Sum the contributions
Now, we sum all these contributions to find the total number of 3s in 1800!:
\[ 600 + 200 + 66 + 22 + 7 + 2 = 897 \]

### Step 5: Find the highest power of 81
Since \( 81 = 3^4 \), we need to determine how many times \( 3^4 \) can fit into the total number of 3s:
\[ \text{Highest power of } 81 = \left\lfloor \frac{897}{4} \right\rfloor = 224 \]

Thus, the highest power of 81 that can divide 1800! is **224**.

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