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# The average of a series of consecutive integers is an even integer

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Intern
Joined: 24 Feb 2017
Posts: 7
The average of a series of consecutive integers is an even integer  [#permalink]

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02 Aug 2017, 12:04
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Difficulty:

65% (hard)

Question Stats:

63% (02:04) correct 38% (02:05) wrong based on 108 sessions

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The average of a series of consecutive integers is an even integer. Which of the following could be true?
I. There is an even number of integers in the series.
II. The smallest number in the series is odd.
III. The product of the smallest and largest number in the series is odd.

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
Re: The average of a series of consecutive integers is an even integer  [#permalink]

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03 Aug 2017, 11:12
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cgalan6 wrote:
The average of a series of consecutive integers is an even integer. Which of the following could be true?
I. There is an even number of integers in the series.
II. The smallest number in the series is odd.
III. The product of the smallest and largest number in the series is odd.

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

Dear cgalan6,

I'm happy to respond.

This is a fabulous question! What is the source? It's great!

Let's think about that first statement:
The average of a series of consecutive integers is an even integer.
The average of a set of consecutive integers is always the median of the set. If we have an even number of consecutive integers, the average & median are non-integers: it's the half-integer between the two middle numbers. It's only if we have an odd number of consecutive integers that the average & median could be an integer at all. The average is the middle number of the set. The two possibilities are
a) even in middle: O E O E O E O or E O E O E O E O E
b) odd in middle: E O E O E O E or O E O E O E O E O
Thus, if we have an even number in the middle, so that the median & average are even, then it could be true that both the first & last are odd, or it could be true that the first & last are even.
Now, we can proceed to the three statements
I. There is an even number of integers in the series.
False. If there are an even number of consecutive integers in the series, then the average wouldn't be an integer at all.

II. The smallest number in the series is odd.
True. This is possible.

III. The product of the smallest and largest number in the series is odd.
True. If the first is odd, the last must also be odd, and their product would be odd. This is possible.

Thus, II and III are possible.

OA = (E)

Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Intern
Joined: 24 Feb 2017
Posts: 7
Re: The average of a series of consecutive integers is an even integer  [#permalink]

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05 Aug 2017, 18:34
mikemcgarry this makes complete sense, but can't a consecutive series of integers also be something like 2,4,6,8 and 3,5,7,9? That's why I was a bit confused because then statement 1 "could" be true. Obviously from the answer choices all three cannot be all true. the And I believe this question is from a kaplan bank. Thanks again.
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
Re: The average of a series of consecutive integers is an even integer  [#permalink]

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07 Aug 2017, 11:18
2
cgalan6 wrote:
mikemcgarry this makes complete sense, but can't a consecutive series of integers also be something like 2,4,6,8 and 3,5,7,9? That's why I was a bit confused because then statement 1 "could" be true. Obviously from the answer choices all three cannot be all true. the And I believe this question is from a kaplan bank. Thanks again.

Dear cgalan6,

I'm happy to respond.

My friend, when any GMAT Quant question talks about "consecutive integers" without adding any other qualification, it ALWAYS means integers that differ by exactly 1 from each of their neighbors—e.g. {32, 33, 34, 35, 36, 37, 38}.

- consecutive even integer, e.g. {-8, -6, -4, -2}
- consecutive odd integers, e.g. {3, 5, 7, 9}
- consecutive multiples of 3, e.g. {-6, -3, 0, 3, 6, 9, 12, 15} etc.
The test may ask about any of those, but it will NEVER just say "consecutive integers" and mean one of those sets.

My friend, precision is the very soul of mathematics, and all mathematical logic depends on precision. On the GMAT Quant, this mathematical precision extends to the words: every single word used in every single prompt on the GMAT Quant is used with extraordinary precision. What I have explained here is the precise meaning of the term "consecutive integers."

Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Intern
Joined: 24 Feb 2017
Posts: 7
Re: The average of a series of consecutive integers is an even integer  [#permalink]

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07 Aug 2017, 12:36
mikemcgarry Thank you oh so very much!!
Intern
Joined: 18 Aug 2017
Posts: 30
GMAT 1: 670 Q49 V33
Re: The average of a series of consecutive integers is an even integer  [#permalink]

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24 Aug 2017, 08:16
If the average of n consecutive interger is an interger, n must be odd.

Therefore 1 can not be true.

Test 2 and 3 with set 3,4,5 => true.

Re: The average of a series of consecutive integers is an even integer   [#permalink] 24 Aug 2017, 08:16
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