How many two-digit numbers are divisible by 3?

Question

Accepted Answer

To determine how many two-digit numbers are divisible by 3, we need to identify the range of two-digit numbers and count those that are divisible by 3.

1. **Range of Two-Digit Numbers**: Two-digit numbers range from 10 to 99.

2. **Finding the First and Last Two-Digit Numbers Divisible by 3**:
   - The smallest two-digit number is 10. The first two-digit number divisible by 3 is 12 (since 12 ÷ 3 = 4).
   - The largest two-digit number is 99. The last two-digit number divisible by 3 is 99 itself (since 99 ÷ 3 = 33).

3. **Counting the Two-Digit Numbers Divisible by 3**:
   - The sequence of two-digit numbers divisible by 3 is an arithmetic sequence: 12, 15, 18, ..., 99.
   - This sequence has a common difference of 3.
   - To find the number of terms in this sequence, we use the formula for the nth term of an arithmetic sequence: 
     $$
     a_n = a_1 + (n-1) \cdot d
     $$
     where $a_n$ is the nth term, $a_1$ is the first term, $d$ is the common difference, and $n$ is the number of terms.
   - Here, $a_1 = 12$, $d = 3$, and $a_n = 99$.
   - Solving for $n$:
     $$
     99 = 12 + (n-1) \cdot 3
     $$
     $$
     99 - 12 = (n-1) \cdot 3
     $$
     $$
     87 = (n-1) \cdot 3
     $$
     $$
     29 = n-1
     $$
     $$
     n = 30
     $$

Therefore, there are 30 two-digit numbers divisible by 3, which confirms that Option 1 is correct.

**Explanation for Incorrect Options**:
- **Option 2 (29)**: This option is incorrect because, as calculated above, there are 30 two-digit numbers divisible by 3, not 29.
- **Option 3 (85)**: This option is incorrect because it significantly overestimates the count. The correct count, as shown, is 30.
- **Option 4 (99)**: This option is incorrect because it also overestimates the count. The correct count is 30, not 99.

In conclusion, the correct answer is Option 1, which states that there are 30 two-digit numbers divisible by 3.