How many numbers between 75 and 500 are divisible by [tex]$7$[/tex]?
Answer (1)
Determine the smallest multiple of 7 greater than 75: 7 × 11 = 77 .
Determine the largest multiple of 7 less than 500: 7 × 71 = 497 .
Calculate the number of multiples of 7 between 77 and 497 inclusive: 71 − 11 + 1 = 61 .
The number of integers between 75 and 500 that are divisible by 7 is 61 .
Explanation
Understanding the Problem We are asked to find how many numbers between 75 and 500 are divisible by 7. This means we are looking for integers x such that 75 < x < 500 and x is a multiple of 7.
Finding the First Multiple First, we need to find the smallest multiple of 7 that is greater than 75. We can do this by dividing 75 by 7 and rounding up to the nearest whole number.
Calculating the First Multiple 7 75 ≈ 10.71 Since we need the smallest integer greater than 10.71, we round up to 11. Thus, the first multiple of 7 greater than 75 is 7 × 11 = 77 .
Finding the Last Multiple Next, we need to find the largest multiple of 7 that is less than 500. We can do this by dividing 500 by 7 and rounding down to the nearest whole number.
Calculating the Last Multiple 7 500 ≈ 71.43 Since we need the largest integer less than 71.43, we round down to 71. Thus, the last multiple of 7 less than 500 is 7 × 71 = 497 .
Counting the Multiples Now we need to find how many multiples of 7 there are between 77 and 497, inclusive. This is the same as finding how many integers there are between 11 and 71, inclusive.
Final Calculation To find the number of integers between 11 and 71, inclusive, we subtract the smaller number from the larger number and add 1: 71 − 11 + 1 = 61 Therefore, there are 61 numbers between 75 and 500 that are divisible by 7.
Final Answer Thus, there are 61 numbers between 75 and 500 that are divisible by 7.
Examples
Imagine you are organizing a conference and need to schedule coffee breaks every 7 minutes. If the conference starts at 75 minutes past the hour and ends at 500 minutes past the hour, you would want to know how many coffee breaks you can schedule. This problem is similar to finding the number of multiples of 7 between 75 and 500. Knowing this helps you plan the conference schedule effectively, ensuring regular breaks for attendees.
