Question

How many many numbers between 75 and 500 are divisible by 7

Answer 1

To find how many numbers between 75 and 500 are divisible by 7, we need to determine the smallest and largest multiples of 7 within that range.

1. **Finding the smallest multiple of 7 greater than or equal to 75**:
- Divide 75 by 7:
\[
75 \div 7 \approx 10.7143
\]
- Taking the ceiling of 10.7143 gives us 11.
- Now, multiply 11 by 7 to get the smallest multiple:
\[
11 \times 7 = 77
\]

2. **Finding the largest multiple of 7 less than or equal to 500**:
- Divide 500 by 7:
\[
500 \div 7 \approx 71.4286
\]
- Taking the floor of 71.4286 gives us 71.
- Now, multiply 71 by 7 to get the largest multiple:
\[
71 \times 7 = 497
\]

3. **Counting the number of multiples of 7 from 77 to 497**:
- The sequence of multiples of 7 can be expressed as:
\[
7n \quad \text{where } n \text{ is an integer}
\]
- For our range, \( n \) starts at 11 (for 77) and ends at 71 (for 497).

4. **Calculating the total count**:
- To find the total non-negative integers \( n \) from 11 to 71, we can use:
\[
\text{Total count} = 71 - 11 + 1 = 61
\]

Thus, there are **61 numbers between 75 and 500 that are divisible by 7**.