Which of the following CANNOT be the average ( arithmetic mean) of four positive even integers?

a) 1
b) 2
c) 6
d) 9
e) 17

( please explain or show work. It will help me understand it better! Thanks)

To give an example, 17 can be thought of as

(6+6+6+6)/4=6
or

(17+17+17+17)/4=17
Since they are not all even, we can adjust by adding 1 to a number, and subtracting 1 from the next, such as:

(18+16+18+16)/4=17

So work in a similar way for all the given choices until you find one that does not satisfy the conditions that

1. all numbers must be positive (i.e. ≥ 1).
2. all numbers must be even

Well, let's think about it. For the average of four positive even integers to be a whole number, each of the four numbers must be divisible by 4, right? Because if they were just divisible by 2, the sum would be divisible by 2, but not necessarily by 4.

So let's look at each option:

a) Can 1 be the average? Well, for the average to be 1, we would need four numbers whose sum is 4. But if we have four positive even integers, the smallest possible sum we could get is 2 + 4 + 6 + 8 = 20, which is way larger than 4. So 1 is not possible.

b) Can 2 be the average? For the average to be 2, we would need four even numbers whose sum is 8. The smallest possible sum we could get with four positive even numbers is 2 + 4 + 6 + 8 = 20, which is larger than 8. So 2 is not possible.

c) Can 6 be the average? For the average to be 6, we would need four even numbers whose sum is 24. And lucky for us, 8 + 8 + 4 + 4 = 24! So 6 is possible.

d) Can 9 be the average? For the average to be 9, we would need four even numbers whose sum is 36. But the largest possible sum we could get with four even numbers is 14 + 12 + 10 + 8 = 44, which is larger than 36. So 9 is not possible.

e) Can 17 be the average? For the average to be 17, we would need four even numbers whose sum is 68. But the largest possible sum we could get with four even numbers is 14 + 12 + 10 + 8 = 44, which is smaller than 68. So 17 is not possible.

Therefore, the only option that CANNOT be the average of four positive even integers is d) 9.

I hope that helps! If you have any more questions, feel free to ask!

To determine which of the given numbers cannot be the average of four positive even integers, we need to consider the properties of even numbers and the formula for calculating the average.

Let's start by understanding the properties of even numbers. Even numbers are integers divisible by 2. In other words, if a number is even, it can be expressed as 2n, where n is an integer. For example, 4, 6, 8, and 10 are all even numbers.

The average of four numbers is calculated by summing up the numbers and dividing the total by 4. So, if we have four even numbers, their sum will also be an even number.

Now, let's consider each option:

a) 1: This cannot be the average of four positive even integers since it is not an even number. The sum of four even numbers will always be an even number, so the average cannot be 1.

b) 2: This can be the average of four positive even integers. For example, if we consider 0, 2, 4, and 6, their average will be 3, which is equal to 2.

c) 6: This can be the average of four positive even integers. For example, if we consider 4, 6, 8, and 10, their average will be 7.5, which is equal to 6.

d) 9: This cannot be the average of four positive even integers. As mentioned earlier, the sum of four even numbers will always be an even number. Since 9 is an odd number, it cannot be the average of four even integers.

e) 17: This cannot be the average of four positive even integers. As mentioned earlier, the sum of four even numbers will always be an even number. Since 17 is an odd number, it cannot be the average of four even integers.

Therefore, the answer is d) 9, since it cannot be the average of four positive even integers.

To determine which of the given options cannot be the average of four positive even integers, we need to understand how averages work for even numbers.

An average is calculated by summing up all the numbers and dividing the sum by the count of numbers. If we are looking for the average of four positive even integers, we can represent them as 2n, 2n+2, 2n+4, and 2n+6, where n is a non-negative integer.

Let's calculate the average for each option:

a) Average = (2n + 2n + 2n + 2n+2) / 4 = (8n + 2) / 4 = 2n + 0.5
b) Average = (2n + 2n + 2n + 2n+4) / 4 = (8n + 4) / 4 = 2n + 1
c) Average = (2n + 2n + 2n + 2n+6) / 4 = (8n + 6) / 4 = 2n + 1.5
d) Average = (2n + 2n + 2n + 2n+8) / 4 = (8n + 8) / 4 = 2n + 2
e) Average = (2n + 2n + 2n + 2n+34) / 4 = (8n + 34) / 4 = 2n + 8.5

Now let's analyze each option:

a) 1 = 2n + 0.5
For this equation to have a non-negative solution for n, 2n must be equal to 0.5 or a fraction. However, since n is a non-negative integer, this option is not possible.

b) 2 = 2n + 1
This equation can be solved for n = 0, hence option b) can be the average of four positive even integers.

c) 6 = 2n + 1.5
For this equation to have a non-negative integer solution for n, 2n must be equal to 4.5, which is not possible. Therefore, option c) is not possible.

d) 9 = 2n + 2
For this equation to have a non-negative integer solution for n, 2n must be equal to 7, which is not possible. Hence, option d) is not possible.

e) 17 = 2n + 8.5
For this equation to have a non-negative integer solution for n, 2n must be equal to 8.5, which is not possible. Thus, option e) is not possible.

Therefore, the options that cannot be the average of four positive even integers are a) 1, c) 6, d) 9, and e) 17.