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GMAT Club Around The World!!! An arithmetic progression is one in whic

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Bunuel
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GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink] New post   31 Dec 2021, 16:00
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An arithmetic progression is one in which each subsequent term is the sum of the preceding number and a constant. If a, b, c, d, e, f, and g are seven integers in arithmetic progression with four of them being odd, which of the following is/are true about the seven integers?

I. The average is not an integer.

II. The median is an even integer.

III. The difference between the average and the median is an even integer.

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III


 


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Bunuel
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Re: GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink] New post   31 Dec 2021, 16:00
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Bunuel wrote:
An arithmetic progression is one in which each subsequent term is the sum of the preceding number and a constant. If a, b, c, d, e, f, and g are seven integers in arithmetic progression with four of them being odd, which of the following is/are true about the seven integers?

I. The average is not an integer.

II. The median is an even integer.

III. The difference between the average and the median is an even integer.

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III



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Re: GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink] New post   31 Dec 2021, 16:59
Bunuel wrote:
An arithmetic progression is one in which each subsequent term is the sum of the preceding number and a constant. If a, b, c, d, e, f, and g are seven integers in arithmetic progression with four of them being odd, which of the following is/are true about the seven integers?

I. The average is not an integer.

II. The median is an even integer.

III. The difference between the average and the median is an even integer.

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III



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Conditions:
a is even and r is even, In this case all numbers will be even but we know 4 numbers are odd. So out.
a is even and r is odd, In this case there will be only 3 odd numbers, we know they are 4. So out.
a is odd and r is even, In this case all numbers will be odd. So out.
a is odd and r is odd, In this case there will be exactly 4 odds.

I. The average is not an integer.
Since its an AP So, average = sum of first and last term/2
a+g/2
odd+odd/2
even/2 It's definitely an integer.

II. The median is an even integer.
Median will be 4th term which is d = even. So true.
Starting from a alternate numbers are odd.

III. The difference between the average and the median is an even integer.
avg = even
median = even
even-even = even True.


So II and III are correct.
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Re: GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink] New post   31 Dec 2021, 22:40
Assume the series as -

3, 4, 5, 6, 7, 8, 9

Total = 42
Average = 6
Median = 6

I. The average is not an integer.

Incorrect

II. The median is an even integer.

Correct

III. The difference between the average and the median is an even integer.

Correct
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Re: GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink] New post   01 Jan 2022, 02:02
Correct Option D

Consider set : {1,2,3,4,5,6,7}
(4 Odd and 3 Even)

I. The average is not an integer.
(1+2+3+4+5+6+7) = 28÷7 = 4 - Reject

II. The median is an even integer.
{1,2,3,4,5,6,7} - Correct


III. The difference between the average and the median is an even integer.
Average :4
Median :4
Average - Median = (4-4) = 0
Zero is considered Even - Correct

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Re: GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink] New post   17 Dec 2023, 05:18
Inference from question stem:
a) As this is an Arithmetic Progression ---> All terms are equally spaced. i.e. an Equally spaced set
For an equally spaced set,
Median = Average
b) No. of integers: 7 i.e. ODD elements
c) Out of seven, 4 no. are ODD :
The only combination possible: O E O E O E O

Statement 1: The average is not an integer
For odd no. of integer elements in an equally spaced set, Median = Average = Integer
Therefore, St-1 is False

Statement 2: The median is an even integer.
As median = average, the 4th element is an Even no from the possible combination.
Therefore, St-2 is True

Statement 3: The difference between the average and the median is an even integer.
Average=Median
This implies, Average-Median = 0
Therefore, St-3 is True

IMO - Option D
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Re: GMAT Club Around The World!!! An arithmetic progression is one in whic [#permalink]
17 Dec 2023, 05:18
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