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# The average of five consecutive integers is an odd number. Which of

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Math Expert
Joined: 02 Sep 2009
Posts: 58364
The average of five consecutive integers is an odd number. Which of  [#permalink]

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13 Mar 2018, 00:02
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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

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Re: The average of five consecutive integers is an odd number. Which of  [#permalink]

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13 Mar 2018, 01:01
As the five consecutive integers should be start and end with odd numbers to satisfy the condition of the question. It is given that the average is odd then it is obvious that sum of integers unit digit needs to be 5 to give and odd average. But if you take 5 integers in which 1st is even then the 5th would also be even which would not give sum which is odd.
For eg: when largest is even
4,5,6,7,8
8+7+6+5+4=30 and the avg of 30 is 6 for 5 consecutive integers. Then this does not satisfy the 1st choice
But when we take example when largest is odd
3,4,5,6,7
3+4+5+6+7= 25
The avg will be 5 which is odd and satisfy the question.
And as we can see the difference between largest and smallest is even for this pair
7-3= 4
Bunuel

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Joined: 03 Nov 2014
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Re: The average of five consecutive integers is an odd number. Which of  [#permalink]

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13 Mar 2018, 01:38

Question says average of first 5 consecutive integers is odd so x+(x+1)+(x+2)+(x+3)+(x+4)/5 & if x is 1 so the answer would be 3 which is odd number.

(1) The largest of the integer is even (NO) because (x+4) is (1+4) which is 5 hence it isnt Even.
(2)The sum of the integers is odd (YES) 1+(1+1)+(1+2)+(1+3)+(1+4)=15 hence its ODD
(3)Difference between smallest & largest integer is an even number (YES) because 1-5=-4 hence its EVEN Number

Please correct me if i am wrong guys! that's how i have came to the conclusion.
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Re: The average of five consecutive integers is an odd number. Which of  [#permalink]

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13 Mar 2018, 01:43
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.

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Re: The average of five consecutive integers is an odd number. Which of   [#permalink] 13 Mar 2018, 01:43
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