What is the smallest 9 digit number containing all the non-zero digits 1,2,..,9, which is divisible by 99?

Options

A. 125364789
B. 124365879
C. 123475689
D. none of these

Answer

To find the smallest 9-digit number containing all the non-zero digits from 1 to 9 that is divisible by 99, we need to ensure that the number meets the divisibility rules for both 9 and 11, since 99 = 9 × 11.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need a 9-digit number that uses each digit from 1 to 9 exactly once and is divisible by 99.

2. **Divisibility by 9**: A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits from 1 to 9 is:
\[ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \]
Since 45 is divisible by 9, any arrangement of these digits will also be divisible by 9.

**Hint**: Check the sum of the digits to verify divisibility by 9.

3. **Divisibility by 11**: A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or divisible by 11.

Let's denote the digits of our number as \(d_1, d_2, d_3, d_4, d_5, d_6, d_7, d_8, d_9\). The sums can be calculated as follows:
- Odd positions: \(d_1 + d_3 + d_5 + d_7 + d_9\)
- Even positions: \(d_2 + d_4 + d_6 + d_8\)

We need to find a combination of the digits that satisfies this condition.

4. **Finding the Smallest Number**: To find the smallest number, we can start arranging the digits from smallest to largest. The smallest arrangement of the digits 1 to 9 is 123456789. However, we need to check if this number is divisible by 11.

5. **Check for Divisibility by 11**:
- For 123456789:
- Odd positions: \(1 + 3 + 5 + 7 + 9 = 25\)
- Even positions: \(2 + 4 + 6 + 8 = 20\)
- Difference: \(25 - 20 = 5\) (not divisible by 11)

We can try different combinations of the digits while keeping the smallest arrangement in mind.

6. **Finding a Valid Combination**: After testing various combinations, we find that the number **123475968** is the smallest arrangement that meets both divisibility rules:
- Odd positions: \(1 + 3 + 7 + 9 + 8 = 28\)
- Even positions: \(2 + 4 + 5 + 6 = 17\)
- Difference: \(28 - 17 = 11\) (divisible by 11)

7. **Conclusion**: Since 123475968 is divisible by both 9 and 11, it is divisible by 99.

### Final Answer:
The smallest 9-digit number containing all the non-zero digits from 1 to 9 that is divisible by 99 is **123475968**.