A natural number N is such that `N = a^2 = b^4 = c^8` , where a, b, c are distinct positive integers, then the least possible value of N is :

Options

A. 729
B. 1000
C. 256
D. None of these

Answer

To solve the problem where a natural number \( N \) is defined as \( N = a^2 = b^4 = c^8 \) with \( a, b, c \) being distinct positive integers, we can follow these steps:

### Step 1: Understand the Relationships
We know that:
- \( N = a^2 \)
- \( N = b^4 \)
- \( N = c^8 \)

This means that \( N \) can be expressed in terms of \( a, b, \) and \( c \) as powers of a common base.

### Step 2: Express \( N \) in Terms of a Common Base
Let \( N = k^{\text{lcm}(2, 4, 8)} \). The least common multiple (LCM) of the exponents 2, 4, and 8 is 8. Therefore, we can express \( N \) as:
\[ N = k^8 \]

From this, we can express \( a, b, \) and \( c \) in terms of \( k \):
- \( a = k^4 \)
- \( b = k^2 \)
- \( c = k \)

### Step 3: Ensure Distinctness
Since \( a, b, c \) must be distinct positive integers, we need to ensure that \( k, k^2, k^4 \) are distinct. This is guaranteed as long as \( k \) is a natural number greater than 1.

### Step 4: Find the Smallest Value of \( k \)
The smallest natural number \( k \) that satisfies this condition is \( k = 2 \):
- \( c = k = 2 \)
- \( b = k^2 = 2^2 = 4 \)
- \( a = k^4 = 2^4 = 16 \)

### Step 5: Calculate \( N \)
Now, substituting \( k = 2 \) into the expression for \( N \):
\[ N = k^8 = 2^8 = 256 \]

### Conclusion
Thus, the least possible value of \( N \) is:
\[ \boxed{256} \]