The arithmetic mean of 17 consecutive integers is an odd num

Vincen 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

The OA is E.

I don't know how to solve this PS question. Experts, may you help me please?

Say the first term is x, thus, the 17 terms would be,

x, (x+1), (x+2), (x+3), (x+4) ............(x+15), & (x+16)

Average of 17 terms = Sum of the 17 terms/17

Sum of the terms = x + (x+1) + (x+2) + (x+3) + (x+4) ............(x+15) + (x+16)
Sum of the terms = 17x + (1 + 2 + 3 + ....... + 16)

Since the series 1 + 2 + 3 + ....... + 16 is equally sapced, its sum = 16*(average of the smallest term and the largest term)

Thus, the sum of the terms = 17x + 16*[(1 + 16)/2] = 17x + 8*17

Thus, the average of 17 terms = (17x + 8*17)/17 = x + 8

We are given that the average of 17 terms is odd, thus, x + 8 = odd. Thus, x must be odd.

Let's take each statement one by one.

I. Largest integer is even: Incorrect. We see that x is odd and the largest term is x + 16. Thus, odd + even = odd.
II. Sum of all integers is odd: Correct. We see that the sum of the terms = 17x + 8*17 = 17(x + 8) = odd*(odd + even) = odd*(odd) = odd
III. Difference between largest and smallest integer is even: Correct. Difference between largest and smallest integer = (x + 16) – x = 16, an even number.

The correct answer: E

Hope this helps!

-Jay

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Vincen wrote: ↑

Mon Oct 23, 2017 7:14 pm

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

The OA is E.

I don't know how to solve this PS question. Experts, may you help me please?

Since the mean (or average) of a set of consecutive integers is equal to the median, we see that the median is also odd. Now let’s analyze the Roman numeral statements.

I. Largest integer is even.

The largest number in terms of the median (or the middle number) is (17 - 1)/2 = 8 more than the median. Since the median is odd and odd + 8 = odd + even = odd, we see that the largest number is odd also. Statement I is not ture.

II. Sum of all integers is odd.

Since sum = average x quantity and here we have average = odd and quantity = 17, we see that Sum = odd x 17 = odd x odd = odd. Statement II is true.

III. Difference between the largest and smallest integers is even.

In I, we see that the largest number is 8 more than the median, i.e., largest number = median + 8. Therefore, the smallest number should be 8 less than the median, i.e., smallest number = median - 8. So we have:

largest number - smallest number = (median + 8) - (median - 8) = 8 + 8 = 16

Statement III is true.

Answer: E