Which of the following can be the average(arithmetic mean ) of four consecutive positive even integers?

a. 2 b.3 c. 4 d. 5 e. 6

To find the average of four consecutive positive even integers, we need to determine the sum of these integers and then divide by 4.

Let's represent the first even integer as x. The second, third, and fourth consecutive even integers can be expressed as x+2, x+4, and x+6, respectively.

So, the sum of these four consecutive even integers is:

x + (x+2) + (x+4) + (x+6) = 4x + 12.

To find the average, we divide the sum by 4:

(4x + 12) / 4 = x + 3.

Now, we have an expression for the average of the four consecutive even integers: x + 3.

To determine which option is a possible average, we need to check which one of the given options is equal to x + 3, where x is a positive even integer.

Let's go through the options one by one:

a. 2: x + 3 = 2. This means x = -1, which is not a positive even integer. So, option a is not valid.

b. 3: x + 3 = 3. This gives us x = 0, which is not a positive integer. So, option b is not valid.

c. 4: x + 3 = 4. This implies x = 1, which is a positive even integer. Therefore, option c is a possible average.

d. 5: x + 3 = 5. This gives us x = 2, which is a positive even integer. Hence, option d is a possible average.

e. 6: x + 3 = 6. This implies x = 3, which is a positive odd integer. Therefore, option e is not valid.

Thus, the possible averages of the four consecutive positive even integers are options c (4) and d (5).