What is the average of all integers between 8 and 74 that are exactly divisible by 7?

Introduction / Context:
This question involves finding the average of all multiples of 7 in a given interval. It tests recognition that those multiples form an arithmetic progression and that the average of an arithmetic progression equals the mean of the first and last term.

Given Data / Assumptions:
- Interval: numbers between 8 and 74 inclusive. - We only consider numbers in this range that are divisible by 7. - We need to find the arithmetic mean of these multiples of 7.

Concept / Approach:
Multiples of 7 form an arithmetic progression with common difference 7. Within a given interval, the first multiple is the smallest number in the interval that is divisible by 7, and the last multiple is the largest such number. For an arithmetic progression, the average is:
average = (first term + last term) / 2. We identify the first and last multiples of 7 in the range from 8 to 74 and then apply this formula directly.

Step-by-Step Solution:
Step 1: Find the first multiple of 7 greater than or equal to 8. The multiples of 7 are 7, 14, 21 and so on. The first multiple greater than 8 is 14. Step 2: Find the last multiple of 7 less than or equal to 74. Multiples near 74 are 70 and 77, but 77 is greater than 74, so the last valid multiple is 70. Step 3: Thus our sequence of terms is 14, 21, 28, ..., 70 with common difference 7. Step 4: Apply the arithmetic progression average formula: average = (first term + last term) / 2. Step 5: Substitute values: average = (14 + 70) / 2. Step 6: Compute sum: 14 + 70 = 84. Step 7: Divide by 2: 84 / 2 = 42. Step 8: Therefore, the required average is 42.

Verification / Alternative check:
We can also compute the count of multiples. The sequence is 14 + 7k for k starting from 0. Solve 14 + 7k = 70 to get k = 8. So there are 9 terms (k = 0 to 8). Sum = number of terms * average = 9 * 42 = 378. If you list them explicitly and add (14 + 21 + 28 + 35 + 42 + 49 + 56 + 63 + 70) you also get 378, confirming the average as 42.

Why Other Options Are Wrong:
Option A (40), B (41), D (43): All are close but do not equal the midpoint between 14 and 70. Option E (36): Too small; it would be the average of a much lower symmetric range than 14 to 70.

Common Pitfalls:
A common mistake is to average 8 and 74 directly instead of focusing on multiples of 7, or to accidentally include 7 or 77 when they fall outside the specified interval. Another error is to sum all multiples manually without recognizing the shortcuts offered by arithmetic progression formulas, which can lead to calculation errors under time pressure.

Final Answer:
The average of all numbers between 8 and 74 that are divisible by 7 is 42.