Kritesh wrote:

The arithmetic mean of 17 consecutive integers is an odd number. Which of the following must be true?

I. Largest integer is even.

II. Sum of all integers is odd.

III. Difference between largest and smallest integer is even.

(A) I

(B) II

(C) III

(D) I, II

(E) II, III

Here is another way to look at this problem.

Given:

17 integers (odd number of integers)

"consecutive" integers, one after another.

Average is odd => SUM / 17 = ODD => SUM = ODD * 17 ==> SUM is odd. Already means II is always true.

Lets look at other choices:

I) Lets say largest number is even, since there are 17 numbers, the smallest number will also be even. Also, there will be 9 even and 8 odd numbers (you can do this with an example, but its pretty intuitive since every other number is odd)

If you want to take an example:

Lets say largest is 18

Smallest will be 2

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 ==> As you can see there are 9 even and 8 odd.

Sum of 9 even numbers => even

Sum of 8 odd numbers => even

Sum of all numbers => even + even = even. So (1) can never be true actually. since the sum is even and we already established sum of numbers should be odd.

This also means that the largest number must be ODD (since it cannot be even). This means smallest number will also be ODD.

Largest - smallest = Odd - Odd = even.

Hence (III) is also true.

Hence I and III are both true.

(E).

Try my

number theory workshop I to learn more about even /odd tips and tricks.

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Abhijit @ Prepitt

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MBA, Columbia Business School (Graduated 2014)

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