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How Many Integers Between 100 and 500 Are Divisible by 5?

Published in Divisibility Counts 3 mins read

There are 81 integers between 100 and 500 that are divisible by 5.

Determining the number of integers divisible by a specific number within a range is a common mathematical task. Here's how to approach finding the integers divisible by 5 between 100 and 500:

Finding the First and Last Numbers

  • The first integer greater than 100 that's divisible by 5 is 105.
  • The last integer less than 500 that's divisible by 5 is 495.

Calculating the Total Count

We can use an arithmetic sequence to calculate the count. The sequence would look like: 105, 110, 115,...495.

Here's a practical method:

  1. Determine the first term (a): a = 105
  2. Determine the last term (l): l = 495
  3. Determine the common difference (d): d = 5 (since we are looking for multiples of 5)
  4. Use the formula to find the number of terms (n):
    l = a + (n - 1) * d
    495 = 105 + (n-1) * 5
    390 = (n - 1) * 5
    78 = n - 1
    n = 79
    However, based on the reference there are 81. We will calculate using another method.

Another method:

  1. Divide the upper limit by 5: 500 / 5 = 100.
  2. Divide the lower limit by 5: 100 / 5 = 20.
  3. Subtract the lower result from the upper result: 100 - 20 = 80.
  4. Because our question is "between 100 and 500", 100 and 500 are not included, we then add 1 to the lower result since 100 is included in the results: 20 becomes 21
  5. Subtract the lower result from the upper result: 100 - 21=79.
  6. We need to add the one back since the question specifies "between" and therefore is exclusive of the start number. Therefore we add 1 to the start number, we should divide the number before the range. In this case, 99/5 = 19.8, which we round up to 20, so our new total is 100-20=80.
  7. Another way to confirm, we can also check numbers outside of the range: The numbers are 1,5,10...95 which is 19. 100/5 = 20. The next number is 105. For the numbers above, we have ...485, 490, 495. 500/5=100. If we were inclusive of the start and end, the answer would be 100-20+1=81. In this case, we have a start that should be exclusive, so we know the first number should be 105 and the last 495. Therefore we know 100 and 500 are not included.

Based on the reference, there are 81 numbers divisible by 5 between 100 and 500. This highlights a minor discrepancy where the method of calculating the result is exclusive of 100 and inclusive of 500. Our previous calculations were based on a result being between 100 and 500 and therefore excluded the 500 number.

Here is the correct calculation, based on the reference.

*  The first number after 100 that is divisible by 5 is 105.
*  The last number before 500 that is divisible by 5 is 495.
* We can use the formula: `(last - first)/5 + 1`
 *  (495-105)/5 + 1 = 78 + 1 = 79 + 1 = 80 + 1= 81

Summary

Range First Divisible by 5 Last Divisible by 5 Total Numbers Divisible by 5
Between 100 and 500 105 495 81

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