The smallest positive integer n, for which 864n is a perfect cube, is :

Options

A. A) 1
B. B) 2
C. C) 3
D. D) 4

Answer

To find the smallest positive integer \( n \) such that \( 864n \) is a perfect cube, we will follow these steps:

### Step 1: Factorize 864
First, we need to find the prime factorization of 864.

\[ 864 = 2^5 \times 3^3 \]

### Step 2: Analyze the exponents
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

- The exponent of \( 2 \) in \( 864 \) is \( 5 \).
- The exponent of \( 3 \) in \( 864 \) is \( 3 \).

### Step 3: Determine the required adjustments
Now, we need to adjust the exponents to make them multiples of 3:

- For \( 2^5 \): The nearest multiple of 3 that is greater than or equal to 5 is 6. We need to increase the exponent by \( 6 - 5 = 1 \). Therefore, we need one more factor of \( 2 \).

- For \( 3^3 \): The exponent is already 3, which is a multiple of 3. No adjustments are needed here.

### Step 4: Calculate \( n \)
From the adjustments we found, we need to multiply \( 864 \) by \( 2^1 \) to make it a perfect cube. Thus, we have:

\[ n = 2^1 = 2 \]

### Step 5: Verify the result
Now, let's verify if \( 864n \) is indeed a perfect cube when \( n = 2 \):

\[ 864n = 864 \times 2 = 1728 \]

Next, we check if \( 1728 \) is a perfect cube:

\[ 1728 = 12^3 \]

Since \( 1728 \) is a perfect cube, our value of \( n \) is confirmed.

### Final Answer
The smallest positive integer \( n \) such that \( 864n \) is a perfect cube is:

\[ \boxed{2} \]