The numbers 1 to 29 are written side by side as follows 1234567891011…..28 29
If the number is divisible by 9, then what is the remainder?

Options

A. 3
B. 1
C. 0
D. None of these

Answer

To solve the problem, we need to find the remainder when the number formed by writing the numbers from 1 to 29 side by side is divided by 9. According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, we will first calculate the sum of the digits of the numbers from 1 to 29 and then find the remainder when this sum is divided by 9.

### Step-by-Step Solution:

1. **Write down the numbers from 1 to 29**:
The numbers are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29.

2. **Calculate the sum of the digits**:
- For single-digit numbers (1 to 9):
\[ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 \]
- For double-digit numbers (10 to 29):
- The digits of 10: \(1 + 0 = 1\)
- The digits of 11: \(1 + 1 = 2\)
- The digits of 12: \(1 + 2 = 3\)
- The digits of 13: \(1 + 3 = 4\)
- The digits of 14: \(1 + 4 = 5\)
- The digits of 15: \(1 + 5 = 6\)
- The digits of 16: \(1 + 6 = 7\)
- The digits of 17: \(1 + 7 = 8\)
- The digits of 18: \(1 + 8 = 9\)
- The digits of 19: \(1 + 9 = 10\)
- The digits of 20: \(2 + 0 = 2\)
- The digits of 21: \(2 + 1 = 3\)
- The digits of 22: \(2 + 2 = 4\)
- The digits of 23: \(2 + 3 = 5\)
- The digits of 24: \(2 + 4 = 6\)
- The digits of 25: \(2 + 5 = 7\)
- The digits of 26: \(2 + 6 = 8\)
- The digits of 27: \(2 + 7 = 9\)
- The digits of 28: \(2 + 8 = 10\)
- The digits of 29: \(2 + 9 = 11\)

- Now, summing these results:
\[ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 435 \]

3. **Find the remainder when the sum is divided by 9**:
- Now we divide 435 by 9:
\[ 435 \div 9 = 48 \quad \text{(quotient)} \]
- Calculate \(48 \times 9 = 432\).
- Now subtract this from 435 to find the remainder:
\[ 435 - 432 = 3 \]

4. **Conclusion**:
The remainder when the number formed by writing the numbers from 1 to 29 side by side is divided by 9 is **3**.