How many integers between 101 and 300 (inclusive) are divisible by 11?

Introduction:
This is a counting problem involving divisibility and a specified range. You are asked to find how many integers between 101 and 300 (inclusive) are divisible by 11. The method uses the idea of the first and last multiples of 11 within the given interval and then counts how many multiples lie between them using a simple formula.

Given Data / Assumptions:

• We consider integers from 101 to 300 inclusive.
• We need numbers N in this range such that 11 divides N exactly (N is a multiple of 11).
• 11, 22, 33, … are the general multiples of 11.

Concept / Approach:
Find the smallest multiple of 11 that is at least 101, and the largest multiple of 11 that is at most 300. If those are 11 * a and 11 * b respectively, then every multiple of 11 in this interval can be written as 11 * k where k runs from a to b. The count is then simply b - a + 1.

Step-by-Step Solution:
Find the first multiple of 11 greater than or equal to 101.11 * 9 = 99 (too small), 11 * 10 = 110 (valid).So the smallest multiple in the range is 110.Find the last multiple of 11 less than or equal to 300.11 * 27 = 297 (valid), 11 * 28 = 308 (too large).So the largest multiple in the range is 297.Now multiples are 11 * k where k goes from 10 to 27 inclusive.Count = 27 - 10 + 1 = 18.

Verification / Alternative check:
You can list them quickly if needed: 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297. Counting them confirms there are 18 multiples of 11 in the interval from 101 to 300 inclusive.

Why Other Options Are Wrong:
17 or 19: These correspond to off-by-one counting errors, such as missing one endpoint or adding an extra multiple beyond 300.20 and 16: These would require either more or fewer multiples than actually exist, not matching the 10 to 27 factor range.

Common Pitfalls:
Not treating the interval as inclusive and accidentally excluding 110 or 297.Miscomputing 300 / 11 and rounding wrongly to find the last multiple.Confusing the count of multiples with the difference of the endpoints without adding 1.

Final Answer:
18