A nine digit number abcdefght is such that a is divisible by 1, ab is divisible by 3 and abcd is divisible by 4 and so on where none of a, b, c, d, … is same andevery digit is a non-zero digit such a number is :

Options

A. 123456789
B. 381654729
C. 126453789
D. 826435791

Answer

To solve the problem of finding a nine-digit number \( abcdefght \) that meets the specified divisibility conditions, we will analyze the options provided step by step.

### Step 1: Understand the Conditions
The conditions we need to satisfy are:
1. \( a \) is divisible by 1 (always true).
2. \( ab \) is divisible by 3.
3. \( abcd \) is divisible by 4.
4. \( abcde \) is divisible by 5.
5. \( abcdef \) is divisible by 6.
6. \( abcdefg \) is divisible by 7.
7. \( abcdefgh \) is divisible by 8.
8. \( abcdefght \) is divisible by 9.

### Step 2: Analyze Each Option
We will check each option against these conditions.

#### Option A: 1, 2, 3, 4, 5, 6, 7, 8, 9
- \( a = 1 \)
- \( ab = 12 \) (divisible by 3)
- \( abcd = 1234 \) (divisible by 4)
- \( abcde = 12345 \) (ends in 5, divisible by 5)
- \( abcdef = 123456 \) (even and sum of digits is 21, divisible by 6)
- \( abcdefg = 1234567 \) (not divisible by 7)
- **Conclusion**: Fails condition for 7.

#### Option B: 3, 8, 1, 6, 5, 4, 7, 2, 9
- \( a = 3 \)
- \( ab = 38 \) (not divisible by 3)
- **Conclusion**: Fails condition for 3.

#### Option C: 1, 2, 6, 4, 5, 3, 7, 8, 9
- \( a = 1 \)
- \( ab = 12 \) (divisible by 3)
- \( abcd = 1264 \) (last two digits 64, divisible by 4)
- \( abcde = 12645 \) (ends in 5, divisible by 5)
- \( abcdef = 126453 \) (even and sum of digits is 21, divisible by 6)
- \( abcdefg = 1264537 \) (not divisible by 7)
- **Conclusion**: Fails condition for 7.

#### Option D: 8, 2, 6, 4, 3, 5, 7, 9, 1
- \( a = 8 \)
- \( ab = 82 \) (not divisible by 3)
- **Conclusion**: Fails condition for 3.

### Step 3: Final Conclusion
After checking all options, we find that none of the options satisfy all the conditions. However, if we had to select the best option based on the conditions met, Option C is the most promising, but it still fails on the divisibility by 7.

### Summary
The nine-digit number \( abcdefght \) that meets the specified conditions does not exist among the provided options.