GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 12 Nov 2019, 03:21 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If a, b, and c are consecutive positive integers and a < b <

Author Message
TAGS:

### Hide Tags

Manager  Joined: 02 Dec 2012
Posts: 173
If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

3
1
4 00:00

Difficulty:   5% (low)

Question Stats: 86% (01:03) correct 14% (01:19) wrong based on 832 sessions

### HideShow timer Statistics

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
Math Expert V
Joined: 02 Sep 2009
Posts: 58973
If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

6
8
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. (Or: the sum of odd number of consecutive integers is ALWAYS divisible by that odd number. )

More:

• If $$k$$ is odd, the sum of $$k$$ consecutive integers is always divisible by $$k$$. Given $$\{9,10,11\}$$, we have $$k=3$$ consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If $$k$$ is even, the sum of $$k$$ consecutive integers is never divisible by $$k$$. Given $$\{9,10,11,12\}$$, we have $$k=4$$ consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of $$k$$ consecutive integers is always divisible by $$k!$$, so by $$k$$ too. Given $$k=4$$ consecutive integers: $$\{3,4,5,6\}$$. The product of 3*4*5*6 is 360, which is divisible by 4!=24.

_________________
##### General Discussion
Intern  Joined: 16 Sep 2010
Posts: 11
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

1
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!
Retired Moderator Joined: 29 Oct 2013
Posts: 248
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

1
1
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

_________________

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15430
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

1
Hi All,

This Roman Numeral question can be solved by either TESTing VALUES or using Number Properties. Here are the various Number Properties involved in this prompt:

We're told that A, B and C are CONSECUTIVE, POSITIVE INTEGERS and that A < B < C. We're asked which of the following MUST be true.

Since the numbers are consecutive, positive integers and A < B < C, we can 'rewrite' the three variables as..
A
B = A+1
C = A+2

I. C - A = 2

Since C = A+2....
C - A =
(A+2) - A =
2
Roman Numeral 1 is always true.

II. ABC is an EVEN integer.

When dealing with 3 consecutive integers, we're guaranteed to have at least one even integer. The options would be:
(even)(odd)(even)
(odd)(even)(odd)

When multiplying ANY integer by an EVEN number, the product is ALWAYS EVEN. Thus Roman Numeral II is always true.

III. (A+B+C)/3 is an integer.

Using the 'rewritten' versions of B and C above, we know that...
(A+B+C) = (A+A+1+A+2) = 3A+3
Since A is an integer, we know that 3A will always be a multiple of 3. Adding a multiple of 3 to 3 (which is also clearly a multiple of 3), we will end up with a sum that is ALWAYS a multiple of 3. Finally, dividing a multiple of 3 by 3 will always give you an integer. Thus, Roman Numeral III is always true.

GMAT assassins aren't born, they're made,
Rich
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58973
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

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.
_________________
Intern  Joined: 16 Sep 2010
Posts: 11
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

awesome. Thanks a lot.
Senior Manager  Joined: 20 Aug 2015
Posts: 384
Location: India
GMAT 1: 760 Q50 V44 Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

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
Target Test Prep Representative G
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2812
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

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.

_________________

# Jeffrey Miller

Jeff@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Intern  B
Joined: 16 May 2017
Posts: 17
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

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.

VP  D
Joined: 09 Mar 2016
Posts: 1229
If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

Bunuel wrote:
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. (Or: the sum of odd number of consecutive integers is ALWAYS divisible by that odd number. )

More:

• If $$k$$ is odd, the sum of $$k$$ consecutive integers is always divisible by $$k$$. Given $$\{9,10,11\}$$, we have $$k=3$$ consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If $$k$$ is even, the sum of $$k$$ consecutive integers is never divisible by $$k$$. Given $$\{9,10,11,12\}$$, we have $$k=4$$ consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of $$k$$ consecutive integers is always divisible by $$k!$$, so by $$k$$ too. Given $$k=4$$ consecutive integers: $$\{3,4,5,6\}$$. The product of 3*4*5*6 is 360, which is divisible by 4!=24.

how can option I be true if i take $$-4<-3<-2$$ (c - a = 2) $$-2-4 = -6$$:? isnt it must be true question ok I got it I must attentively read the question Non-Human User Joined: 09 Sep 2013
Posts: 13559
Re: If a, b, and c are consecutive positive integers and a < b <  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: If a, b, and c are consecutive positive integers and a < b <   [#permalink] 04 Sep 2019, 12:10
Display posts from previous: Sort by

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