What is the total number of integers between 100 and 200 divisible by 3?

The total number of integers between 100 and 200 that are divisible by 3 is 33.

Understanding the Calculation

To find the count of integers divisible by a specific number within a given range, a common and efficient method involves using integer division. This approach helps determine how many multiples of a number exist up to a certain point, and by subtracting these counts, we can isolate the numbers within a desired range.

The problem asks for integers between 100 and 200 that are divisible by 3. This generally implies numbers strictly greater than 100 and strictly less than 200 (i.e., from 101 to 199). However, the method applied in the Beat The GMAT Forum reference effectively counts numbers greater than 100 up to and including 200. Since neither 100 nor 200 are divisible by 3, the result remains consistent whether the endpoints are strictly excluded or considered as part of the bounds for the counting method.

Steps to determine the count:

1. Count multiples up to the upper limit: Divide the upper limit (200) by 3 and take the floor (the largest integer less than or equal to the result). This gives the number of multiples of 3 from 1 up to 200.

   • 200 / 3 = 66 with a remainder of 2. So, there are 66 integers divisible by 3 up to 200 (i.e., 3, 6, ..., 198).

2. Count multiples up to the lower limit: Divide the lower limit (100) by 3 and take the floor. This gives the number of multiples of 3 from 1 up to 100.

   • 100 / 3 = 33 with a remainder of 1. So, there are 33 integers divisible by 3 up to 100 (i.e., 3, 6, ..., 99).

3. Subtract to find the range count: Subtract the count from the lower limit from the count of the upper limit. This difference represents the total number of integers divisible by 3 that fall after 100 and up to 200.

   • 66 - 33 = 33

This calculation, as detailed in the provided reference, confirms that there are 33 integers divisible by 3 between 100 and 200. The first such integer is 102 (3 × 34) and the last is 198 (3 × 66).

Summary Table

Here's a concise overview of the calculation: