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# If a, b, and c are consecutive positive integers and a < b <

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If a, b, and c are consecutive positive integers and a < b < [#permalink]

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29 Dec 2012, 06:44
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If a, b, and c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer.
III. (a + b + c)/3 is an integer.

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III
[Reveal] Spoiler: OA

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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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29 Dec 2012, 06:49
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If a, b, and c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer.
III. (a + b + c)/3 is an integer.

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III

Since a, b, and c are consecutive positive integers and a < b < c, then c = a + 2, from which it follows that c - a = 2. So, I is true.

Next, out of 3 consecutive integers at least 1 must be even, thus abc=even. II is true.

Finally, since b = a + 1, and c = a + 2, then (a + b + c)/3 = (a + a + 1 + a + 2)/3 = a + 1 = integer. III is true as well.

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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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22 Jun 2013, 11:32
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Hey Bunuel, I know I'm a few years late to this but I have a general question about consecutive integers.
According to your explanation, consecutive integers are always 1 apart. However in Sackmann's Total GMAT Math, he defines 'consecutive integers' as any set of integers that are EVENLY SPACED. I'm a little confused here. What's the correct way to think about them?
Thanks!

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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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22 Jun 2013, 11:37
tricialin wrote:
Hey Bunuel, I know I'm a few years late to this but I have a general question about consecutive integers.
According to your explanation, consecutive integers are always 1 apart. However in Sackmann's Total GMAT Math, he defines 'consecutive integers' as any set of integers that are EVENLY SPACED. I'm a little confused here. What's the correct way to think about them?
Thanks!

When we see "consecutive integers" it ALWAYS means integers that follow each other in order with common difference of 1: ... x-3, x-2, x-1, x, x+1, x+2, ....

For example:

-7, -6, -5 are consecutive integers.

2, 4, 6 ARE NOT consecutive integers, they are consecutive even integers.

3, 5, 7 ARE NOT consecutive integers, they are consecutive odd integers.

So, not all evenly spaced sets represent consecutive integers.
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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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22 Jun 2013, 11:44
awesome. Thanks a lot.

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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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28 May 2014, 09:54
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If a, b, and c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2 #True: Since a,b,c are consecutive integers a and c must be 2 integers apart
II. abc is an even integer. #True: For any 3 consecutive integers a,b,c the product has to be divisible by 3! i.e. 6 --> it is even
III. (a + b + c)/3 is an integer. #True: This is a mean of the series. That is this has to be the middle no. b which as per the question stem is an integer

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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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23 Sep 2015, 20:21
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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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24 Sep 2015, 01:00
Assume the following variables to make the calculations simpler:
3 consecutive integers: x-1, x, x+1
3 consecutive even/odd integers: x-2, x, x+2

Although, this question can be solved without the information, still you should keep it in mind if you encounter questions involving consecutive integers/even integers/odd integers etc.

If a, b, and c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2. By our assumption, (a, b, c) are x-1, 1, x+1.
c - a = (x+1) - (x-1) = 2
Correct

II. abc is an even integer.
2 or more consecutive integers will always be even as every alternate number is even
Correct

III. (a + b + c)/3 is an integer.
((x+1) + x + (x-1))/3 = 3x/3 =x
And x is an integer.
Correct.

Hence option E: I, II, and III
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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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15 Jul 2016, 04:49
If a, b, and c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer.
III. (a + b + c)/3 is an integer.

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III

The easiest way to solve the problem is to plug in some real numbers for a, b, and c. Since we know they are consecutive and we know that a < b < c, we can say:

a = 1

b = 2

c = 3

or

a = 2

b = 3

c = 4

It is good to test two cases because in our first case we start with an odd integer and in the second case we start with an even integer.

Let’s use these values in each Roman numeral answer choice. Remember we need to determine which answer must be true, meaning in all circumstances.

I. c – a = 2

Case #1

3 – 1 = 2

Case #2

4 - 2 = 2

I must be true.

II. abc is an even integer.

Case #1

1 x 2 x 3 = 6

Case #2

2 x 3 x 4 = 24

II must be true.

III. (a + b + c)/3 is an integer.

Case #1

(1 + 2 + 3)/3 = 6/3 = 2

Case #2

(2 + 3 + 4)/3 = 9/3 = 3

III must be true.

I, II, and III are all true.

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Re: If a, b, and c are consecutive positive integers and a < b < [#permalink]

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18 May 2017, 11:52
Here we have a, b, and c are consecutive integers, a<b<c =>

n, n+1, n+2 or
a, b = a+1, c = b +1, c = a+1 + 1 = a + 2 .

1. c=a+2 => sufficient

2. a*b*c => we have 2 variations here: - odd * even * odd or even *odd *even => the result is always going to be even, since we have even number in multiplication => sufficient

3. (a + b + c)/3 = (a + a + 1 + a + 3)/3 = a + 1 – always an integer => sufficient.

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Re: If a, b, and c are consecutive positive integers and a < b <   [#permalink] 18 May 2017, 11:52
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