How Many Integers Between 200 and 500 Are Divisible by 8?

There are 37 or 38 integers between 200 and 500 that are divisible by 8, depending on whether the boundaries (200 and 500) are included. The discrepancy arises from different interpretations of "between."

Understanding the Calculation

To find the number of integers divisible by 8 within a given range, we can use the following method:

1. Find the first multiple: Determine the smallest multiple of 8 greater than or equal to 200. This is 208 (8 x 26).

2. Find the last multiple: Determine the largest multiple of 8 less than or equal to 500. This is 496 (8 x 62).

3. Calculate the number of multiples: The multiples form an arithmetic sequence with a common difference of 8. We can use the formula for the nth term of an arithmetic sequence: a_n = a₁ + (n-1)d, where a_n is the nth term, a₁ is the first term, n is the number of terms, and d is the common difference.

   In this case:

   • a_n = 496
   • a₁ = 208
   • d = 8

   Solving for n:

   496 = 208 + (n-1)8
   288 = (n-1)8
   36 = n-1
   n = 37

Therefore, there are 37 multiples of 8 between 200 and 500, excluding 200 and 500 themselves. If we include 200 and 500 in the range, some sources would consider the integer 200 divisible by 8 and then 38 integers are the solution. This difference in answers illustrates the importance of clearly defining the range boundaries.

Different Interpretations & Results

• Interpretation 1 (Exclusive): If "between 200 and 500" means excluding 200 and 500, then there are 37 integers divisible by 8. This aligns with several online resources cited that explicitly used this approach.

• Interpretation 2 (Inclusive): If "between 200 and 500" includes 200 and 500, then there are 38 integers divisible by 8. This is mentioned in other sources that interpreted the given question using this inclusive approach.

The provided references show a disparity in the answer, highlighting the ambiguity in the original question's phrasing.