The average (arithmetic mean) of 17 consecutive integers is an odd num

Question

The average (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 only
(B) II only
(C) III only
(D) I and II only
(E) II and III only

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EgmatQuantExpert · Answer

Solution

Given  
 • Average (arithmetic mean) of 17 consecutive integers is an odd number

To find  
 • Correct statements

Approach and Working out

Let the consecutive integers are a, a+1, a+2, a+3, ……., a+16.
 • Sum of all the integers = 17a + (1+2+3+ .. +16) = 17a + 16*17/2 = 17a + 8*17= 17(a+8)
 o Average = 17(a + 8)/17 = a + 8 =odd
  So, a = Odd
 Thus, first term is odd.

Statement 1:
 • a + 16 = odd + even = odd
• Not true .

Statement 2:
 • Sum = 17(a+8) = odd × odd = odd
• True

Statement 3:
 • Largest integer – Smallest integer = a+ 16 – a = 16= Even
• True

Thus, option E is the correct answer.
   Correct Answer: Option E

exc4libur · Answer

{1,2,3,4,5}: mean = 3 (odd); rng=5-1=4 (even); sum=3*5=15 (odd)
{2,3,4,5,6}: mean = 4 (even); rng=6-2=4 (even); sum=4*5=20 (even)

Ans (E)

Abhishek009 · Answer

Sum of first n numbers is  1+ 2+ ... + n = \frac{n(n+1) }{ 2}

Now, sum of first 17 numbers will be  \frac{17*18}{2} = 153  and its average will be  \frac{153}{17} = 9  (Odd)

1. Largest number is Odd (17)
2. Sum of all integers is Odd (153)
3. Difference between largest and smallest integer is even. ( 17 - 1 = 16)

Hence, only (E) holds true, Answer must be (E)

satya2029 · Answer

a9 is odd
For this a1 is odd and a17 is odd
so largest integer is  odd 
For consecutive integers 
Sum of the terms= AM*no. of terms
and AM= (First +Last)/2=odd (given)
Sum of the terms= odd* odd= odd 
a1 is odd and a17 is odd
Difference=a17-a1=  Even 
E:)

ScottTargetTestPrep · Answer

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

sujoykrdatta · Answer

Question:  
The average (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 only
(B) II only
(C) III only
(D) I and II only
(E) II and III only

Solution:  
We know that: The average (arithmetic mean) of 17 consecutive integers is an odd number.

Consecutive integers refer to integers having a common difference of 1, for example: 11, 12, 13, 14, .. 
In such a scenario, the arithmetic mean (average) is  always  equal to the median

Thus, for 17 integers, the median is the    th  term i.e. the 9th term
Since the above median (or mean) is an odd number, we have:

The 9th number is odd (hence: 8th is even, 7th is odd ... and the 1st number is odd)
Similarly, the 17th term is 8 more than the 9th term i.e. 8 more than an odd number; hence is odd

Let us look at the statements: 
I) The largest number is the 17th term, which is odd
II) Sum of all numbers = 17 x (Mean) = 17 x (odd number), which is odd
III) Difference between the largest and smallest numbers = Odd - Odd = Even

Thus, statements II and III are correct

Answer E

KarishmaB · Answer

The average (arithmetic mean) of 17 consecutive integers is an odd number - This means the middle number (9th) is odd.
So the 17 numbers are made up of 8 numbers before and after the middle number. e.g. if the middle number (average) is 9, the numbers would be 1 to 17 - 1 to 8 before 9 and 10 to 17 after 9.

I. Largest integer is even.
Largest integer here is 17. Not even.

II. Sum of all integers is odd.
We have 8 even integers and 9 odd integers. Since we have an odd number of odd integers, the sum will be odd.

III. Difference between largest and smallest integer is even.
Smallest and the largest integers both will be odd. So their difference will be even.

Answer (E)

bumpbot · Answer

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The average (arithmetic mean) of 17 consecutive integers is an odd num

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