In each of the following questions, four numbers are given. Out of these three are alike in some way but rest one is different. Choose the one which is different from the other.

Options

A. 126
B. 217
C. 345
D. 513

Answer

To solve the problem, we need to identify the number that is different from the others based on a specific pattern. Let's analyze the given numbers step by step.

### Step 1: List the given numbers
The numbers provided are:
1. 126
2. 217
3. 345
4. 513

### Step 2: Analyze each number
We will check if each number can be expressed as a cube of an integer plus a certain value.

- **Option 1: 126**
- We check if it can be expressed as \( n^3 + k \).
- \( 5^3 = 125 \)
- \( 125 + 1 = 126 \) (So, \( 126 = 5^3 + 1 \))

- **Option 2: 217**
- Check if it can be expressed similarly.
- \( 6^3 = 216 \)
- \( 216 + 1 = 217 \) (So, \( 217 = 6^3 + 1 \))

- **Option 3: 345**
- Check if it can be expressed similarly.
- \( 7^3 = 343 \)
- \( 343 + 2 = 345 \) (So, \( 345 = 7^3 + 2 \))

- **Option 4: 513**
- Check if it can be expressed similarly.
- \( 8^3 = 512 \)
- \( 512 + 1 = 513 \) (So, \( 513 = 8^3 + 1 \))

### Step 3: Identify the pattern
From the analysis:
- Options 1, 2, and 4 can be expressed as \( n^3 + 1 \).
- Option 3 is expressed as \( n^3 + 2 \).

### Step 4: Conclusion
Since options 1, 2, and 4 follow the same pattern of being \( n^3 + 1 \) while option 3 does not, we conclude that:

**The number that is different is 345.**

### Final Answer
The correct answer is option 3: **345**.

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