What is the average (arithmetic mean) of all integers between 100 and 200 inclusive that are exactly divisible by 13?

Introduction / Context:
This question deals with averages of numbers in an arithmetic progression. You are asked to find the average of all integers between 100 and 200 that are divisible by 13. Because these numbers form a regular sequence with a fixed difference, there is a quick way to find the average without listing and adding all of them.

Given Data / Assumptions:
- We consider integers from 100 to 200 inclusive.
- We only include those numbers that are divisible by 13.
- We must find the average (arithmetic mean) of these selected numbers.

Concept / Approach:
All integers divisible by 13 form an arithmetic progression with common difference 13. The first multiple of 13 in the given range and the last multiple of 13 in the range determine the sequence. For any arithmetic progression, the average equals the midpoint of the first and last term, that is (first + last) / 2. So our main task is to identify the first and last multiples of 13 between 100 and 200 and then use this formula.

Step-by-Step Solution:
Step 1: Find the smallest multiple of 13 that is at least 100.Step 2: 13 * 7 = 91 (too small), 13 * 8 = 104, which lies within 100 to 200, so the first term is 104.Step 3: Find the largest multiple of 13 that is at most 200.Step 4: 13 * 15 = 195, and 13 * 16 = 208 (too large), so the last term in the range is 195.Step 5: The numbers divisible by 13 in the range form the sequence 104, 117, 130, ..., 195 with common difference 13.Step 6: Average of an arithmetic progression = (first term + last term) / 2.Step 7: Average = (104 + 195) / 2 = 299 / 2 = 149.5.Step 8: Thus, the required average is 149.5.

Verification / Alternative check:
You can list all multiples of 13 between 100 and 200 to confirm: 104, 117, 130, 143, 156, 169, 182, 195. There are 8 terms. Pairing from the ends, 104 + 195 = 299, 117 + 182 = 299, 130 + 169 = 299, 143 + 156 = 299. Each pair sums to 299, so the average of each pair is 299 / 2 = 149.5. Therefore the average of all 8 numbers is also 149.5, which agrees with our formula based method.

Why Other Options Are Wrong:
Values like 147.5, 145.5 or 143.5 correspond to midpoints of different ranges of numbers and do not match the specific first and last multiple of 13 in the 100 to 200 interval. The value 151.5 is higher than the midpoint of 104 and 195 and would correspond to a different set of end values. Only 149.5 is exactly the average of the true first and last multiples of 13 in the given range.

Common Pitfalls:
Some students forget to include the boundary values correctly or misidentify the first and last multiples of 13. Others attempt to compute the sum of all numbers individually, which is slower and prone to arithmetic mistakes. Remember that in any evenly spaced sequence, the average is simply the midpoint between the first and last term.

Final Answer:
The average of the numbers between 100 and 200 that are divisible by 13 is 149.5.