Bunuel wrote:

The average of five consecutive integers is an odd number. Which of the following must be true?

I. The largest of the integers is even

II. The sum of the integers is odd

III. The difference between the largest and the smalles integers is an even number

A. I only

B. II only

C. III only

D. I and II only

E. II and III only

In addition to the number-property based Logical approach given above, we'll also show a number-based, Alternative approach.

Writing our 5 numbers as x, x+1, x+2, x+3, x+4 then their sum is 5x + 10 and their average is x + 2.

Let's choose the smallest possible x so that x + 2 is odd.

In this case, x = 1 and our numbers are 1,2,3,4,5.

Then I. is false, II is 5 + 10 = 15 is true and III is 5-1=4 is true.

Trying the next smallest x such that x + 2 is odd, x = 3 gives 3,4,5,6,7.

Then I is false, II is 5*3+10 = 25 and III is 7-3=4.

At this point we should feel comfortable marking (E).

Taking a step back, we can realize that x + 2 is odd means that x is odd and therefore only II and III can be true.

(E) is our answer.

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