How many numbers are divisible by 3 in the set of numbers 300, 301, 302, ……., 499, 500 ?

Options

A. 200
B. 66
C. 67
D. None of these

Answer

To find how many numbers are divisible by 3 in the set of numbers from 300 to 500, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the range**: We are looking at the numbers from 300 to 500, inclusive.

2. **Find the first number divisible by 3**:
- The first number in our range is 300.
- Since 300 is divisible by 3 (300 ÷ 3 = 100), the first number is 300.

3. **Find the last number divisible by 3**:
- The last number in our range is 500.
- We need to find the largest number less than or equal to 500 that is divisible by 3.
- Dividing 500 by 3 gives us approximately 166.67.
- The largest integer less than or equal to this is 166, and multiplying back gives us 166 × 3 = 498.
- Thus, the last number divisible by 3 in our range is 498.

4. **Identify the sequence of numbers divisible by 3**:
- The numbers divisible by 3 from 300 to 498 form an arithmetic progression (AP) where:
- First term (A) = 300
- Common difference (D) = 3
- Last term (L) = 498

5. **Use the formula for the nth term of an AP**:
- The formula for the nth term of an AP is given by:
\[ L = A + (N - 1) \cdot D \]
- Substituting the known values:
\[ 498 = 300 + (N - 1) \cdot 3 \]

6. **Solve for N**:
- Rearranging the equation:
\[ 498 - 300 = (N - 1) \cdot 3 \]
\[ 198 = (N - 1) \cdot 3 \]
\[ N - 1 = \frac{198}{3} \]
\[ N - 1 = 66 \]
\[ N = 66 + 1 = 67 \]

7. **Conclusion**:
- Therefore, the total number of integers between 300 and 500 that are divisible by 3 is **67**.