How many two digit natural numbers are divisible by 3 ?

Question

Accepted Answer

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

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

2. **Finding Numbers Divisible by 3**:
   - The smallest two-digit number divisible by 3 is 12 (since 12 ÷ 3 = 4).
   - The largest two-digit number divisible by 3 is 99 (since 99 ÷ 3 = 33).

3. **Counting the Numbers**:
   - We need to find how many numbers are there from 12 to 99 that are divisible by 3.
   - This forms an arithmetic sequence where the first term $a_1$ is 12, the last term $a_n$ is 99, and the common difference $d$ is 3.

4. **Using the Formula for the nth Term of an Arithmetic Sequence**:
   - The nth term of an arithmetic sequence is given by: $a_n = a_1 + (n-1) \times d$.
   - Plugging in the values: $99 = 12 + (n-1) \times 3$.
   - Solving for $n$:
     $$
     99 = 12 + 3(n-1)
     $$
     $$
     99 = 12 + 3n - 3
     $$
     $$
     99 = 9 + 3n
     $$
     $$
     90 = 3n
     $$
     $$
     n = 30
     $$

Therefore, there are 30 two-digit natural numbers that are divisible by 3, which supports Option 2 as the correct answer.

**Explanation for Incorrect Options**:
- **Option 1 (60)**: This option is incorrect because, as calculated, there are only 30 two-digit numbers divisible by 3, not 60.
- **Option 3 (50)**: This option is incorrect because the count of numbers divisible by 3 is 30, not 50.
- **Option 4 (20)**: This option is incorrect because the actual count is 30, which is greater than 20.

In conclusion, the correct answer is Option 2, as there are exactly 30 two-digit natural numbers divisible by 3.