How many natural numbers between 1 and 500 are divisible by each of the numbers 3, 5 and 7?

Question

Accepted Answer

To determine how many natural numbers between 1 and 500 are divisible by each of the numbers 3, 5, and 7, we need to find the numbers that are divisible by the least common multiple (LCM) of these three numbers.

1. **Calculate the LCM of 3, 5, and 7:**
   - The prime factorization of 3 is $3^1$.
   - The prime factorization of 5 is $5^1$.
   - The prime factorization of 7 is $7^1$.
   - The LCM is found by taking the highest power of each prime that appears in the factorizations: $3^1 \times 5^1 \times 7^1 = 105$.

2. **Find the numbers between 1 and 500 that are divisible by 105:**
   - We need to find the multiples of 105 that are less than or equal to 500.
   - The multiples of 105 are: 105, 210, 315, 420, 525, etc.
   - However, since we are only interested in numbers up to 500, we consider only 105, 210, 315, and 420.

3. **Count the multiples:**
   - There are 4 multiples of 105 within the range from 1 to 500: 105, 210, 315, and 420.

Therefore, the number of natural numbers between 1 and 500 that are divisible by each of the numbers 3, 5, and 7 is 4. This supports the given correct answer, which is Option 2.

**Explanation of why other options are incorrect:**

- **Option 1 (3):** This option is incorrect because, as calculated, there are 4 numbers (105, 210, 315, and 420) that meet the criteria, not 3.
  
- **Option 3 (5):** This option is incorrect because there are only 4 numbers that are divisible by 105 within the specified range, not 5.
  
- **Option 4 (6):** This option is incorrect because there are only 4 numbers that satisfy the condition, not 6.

In conclusion, the correct answer is indeed Option 2, as there are exactly 4 natural numbers between 1 and 500 that are divisible by 3, 5, and 7.