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.