(i) How many number of two digits are divisible by 3 ?
(ii) How many numbers of three digits are divisible by 9?

Answer

To solve the questions step by step, we will break down each part clearly.

### Part (i): How many two-digit numbers are divisible by 3?

1. **Identify the range of two-digit numbers**:
The smallest two-digit number is 10 and the largest is 99.

2. **Find the first two-digit number divisible by 3**:
- Start from 10.
- Check if 10 is divisible by 3: \(10 \div 3 = 3.33\) (not divisible).
- Check the next number, 11: \(11 \div 3 = 3.67\) (not divisible).
- Check 12: \(12 \div 3 = 4\) (divisible).
- So, the first two-digit number divisible by 3 is **12**.

3. **Find the last two-digit number divisible by 3**:
- Start from 99.
- Check if 99 is divisible by 3: \(99 \div 3 = 33\) (divisible).
- So, the last two-digit number divisible by 3 is **99**.

4. **Form an arithmetic progression (AP)**:
- The first term \(a = 12\), the last term \(l = 99\), and the common difference \(d = 3\).
- The sequence of two-digit numbers divisible by 3 is: 12, 15, 18, ..., 99.

5. **Use the formula for the nth term of an AP**:
- The nth term of an AP is given by:
\[ T_n = a + (n-1)d \]
- Set \(T_n = 99\):
\[ 99 = 12 + (n-1) \cdot 3 \]
- Rearranging gives:
\[ 99 - 12 = (n-1) \cdot 3 \implies 87 = (n-1) \cdot 3 \]
- Dividing both sides by 3:
\[ n - 1 = 29 \implies n = 30 \]

6. **Conclusion**:
- There are **30 two-digit numbers divisible by 3**.

### Part (ii): How many three-digit numbers are divisible by 9?

1. **Identify the range of three-digit numbers**:
The smallest three-digit number is 100 and the largest is 999.

2. **Find the first three-digit number divisible by 9**:
- Start from 100.
- Check if 100 is divisible by 9: \(100 \div 9 \approx 11.11\) (not divisible).
- Check the next number, 101: \(101 \div 9 \approx 11.22\) (not divisible).
- Check 102: \(102 \div 9 = 11.33\) (not divisible).
- Check 108: \(108 \div 9 = 12\) (divisible).
- So, the first three-digit number divisible by 9 is **108**.

3. **Find the last three-digit number divisible by 9**:
- Start from 999.
- Check if 999 is divisible by 9: \(999 \div 9 = 111\) (divisible).
- So, the last three-digit number divisible by 9 is **999**.

4. **Form an arithmetic progression (AP)**:
- The first term \(a = 108\), the last term \(l = 999\), and the common difference \(d = 9\).
- The sequence of three-digit numbers divisible by 9 is: 108, 117, 126, ..., 999.

5. **Use the formula for the nth term of an AP**:
- Set \(T_n = 999\):
\[ 999 = 108 + (n-1) \cdot 9 \]
- Rearranging gives:
\[ 999 - 108 = (n-1) \cdot 9 \implies 891 = (n-1) \cdot 9 \]
- Dividing both sides by 9:
\[ n - 1 = 99 \implies n = 100 \]

6. **Conclusion**:
- There are **100 three-digit numbers divisible by 9**.