If a, b, c be all positive integers, then the least positive value of `a^(3) + b^(3) + c^(3) - 3abc ` is

Options

A. (A) 0
B. (B) 2
C. (C) 4
D. (D) 3

Answer

To find the least positive value of the expression \( a^3 + b^3 + c^3 - 3abc \) for positive integers \( a, b, c \), we can use a well-known identity in algebra.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \( a^3 + b^3 + c^3 - 3abc \) can be rewritten using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
This identity indicates that the expression equals zero when \( a = b = c \).

2. **Setting Values for \( a, b, c \)**:
Since we are looking for the least positive value, we can start by setting \( a = b = c = 1 \):
\[ a = 1, \quad b = 1, \quad c = 1 \]

3. **Calculating the Expression**:
Substitute \( a, b, c \) into the expression:
\[ 1^3 + 1^3 + 1^3 - 3 \cdot 1 \cdot 1 \cdot 1 = 1 + 1 + 1 - 3 = 3 - 3 = 0 \]

4. **Finding Positive Integer Values**:
Since we need the least positive value, we can try the next set of equal integers, \( a = b = c = 2 \):
\[ 2^3 + 2^3 + 2^3 - 3 \cdot 2 \cdot 2 \cdot 2 = 8 + 8 + 8 - 24 = 24 - 24 = 0 \]
This still results in zero.

5. **Trying Different Combinations**:
Next, we can try \( a = 1, b = 1, c = 2 \):
\[ 1^3 + 1^3 + 2^3 - 3 \cdot 1 \cdot 1 \cdot 2 = 1 + 1 + 8 - 6 = 10 - 6 = 4 \]

6. **Conclusion**:
The least positive value of \( a^3 + b^3 + c^3 - 3abc \) for positive integers \( a, b, c \) is \( 4 \).

### Final Answer:
The least positive value of \( a^3 + b^3 + c^3 - 3abc \) is \( 4 \).