The smallest possible number that can be expressed as the sum of cube of two natural numbers in two different combinations.

Options

A. a.1000
B. b.1728
C. c.1729
D. d.none of these

Answer

To find the smallest possible number that can be expressed as the sum of the cubes of two natural numbers in two different combinations, we can follow these steps:

### Step 1: Understand the problem
We need to find a number \( N \) such that there exist two pairs of natural numbers \( (a, b) \) and \( (c, d) \) such that:
\[ N = a^3 + b^3 = c^3 + d^3 \]
where \( (a, b) \) and \( (c, d) \) are different pairs.

### Step 2: Start calculating cubes of natural numbers
We will calculate the cubes of the first few natural numbers:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)
- \( 10^3 = 1000 \)

### Step 3: Find combinations of cubes
Now we need to find combinations of these cubes that yield the same sum. We can start checking pairs systematically.

1. **First combination**:
- \( 1^3 + 12^3 = 1 + 1728 = 1729 \)

2. **Second combination**:
- \( 9^3 + 10^3 = 729 + 1000 = 1729 \)

### Step 4: Verify the combinations
We have found:
- \( 1^3 + 12^3 = 1729 \)
- \( 9^3 + 10^3 = 1729 \)

Both combinations yield the same result, confirming that \( 1729 \) can be expressed as the sum of cubes of two different pairs of natural numbers.

### Conclusion
Thus, the smallest possible number that can be expressed as the sum of the cubes of two natural numbers in two different combinations is:
\[ \boxed{1729} \]

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