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

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The average (arithmetic mean) of 17 consecutive integers is an odd num  [#permalink]

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New post 15 Nov 2019, 01:47
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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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Re: The average (arithmetic mean) of 17 consecutive integers is an odd num  [#permalink]

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New post 15 Nov 2019, 04:10

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
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Re: The average (arithmetic mean) of 17 consecutive integers is an odd num  [#permalink]

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New post 19 Nov 2019, 04:49
Bunuel wrote:
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


{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)
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Re: The average (arithmetic mean) of 17 consecutive integers is an odd num  [#permalink]

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New post 19 Nov 2019, 11:32
Bunuel wrote:
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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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)
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Re: The average (arithmetic mean) of 17 consecutive integers is an odd num  [#permalink]

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New post 22 Nov 2019, 07:20
Bunuel wrote:
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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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:)
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Re: The average (arithmetic mean) of 17 consecutive integers is an odd num   [#permalink] 22 Nov 2019, 07:20
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