consecutive odd integers

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prachi18oct: Master | Next Rank: 500 Posts; Posts: 269; Joined: Sun Apr 27, 2014 10:33 pmFollowed by: 5 members; Thanked: 8 times

by prachi18oct » Wed Aug 20, 2014 6:41 am

If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

x + y is even
is an integer
xz is even

I only
II only
III only
I and II only
I, II, and III

I eliminated option D as I considered -1,1,3 as one of the possibilities and hence -1+1 = 0 which is neutral.Please suggest if we should consider this possibility.

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Source: Beat The GMAT — Problem Solving | Wed Aug 20, 2014 6:41 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Aug 20, 2014 6:48 am

prachi18oct wrote:If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

x + y is even
is an integer
xz is even

I only
II only
III only
I and II only
I, II, and III

I eliminated option D as I considered -1,1,3 as one of the possibilities and hence -1+1 = 0 which is neutral.Please suggest if we should consider this possibility.

You're missing part of the question (see above)

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by Brent@GMATPrepNow » Wed Aug 20, 2014 7:26 am

I found the original question. It reads as follows:

If x, y, and z are consecutive odd integers, with x < y < z, then which of the following MUST be true?

I. x + y is even
II. (x + z)/y is an integer
III. xz is even

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

The key word here is MUST

I. x + y is even
Since x and y are both ODD, x + y = ODD + ODD = EVEN
So, statement I is true

II. (x + z)/y is an integer
Must this be true?
Since x, y and z are consecutive odd integers, we know that y is 2 greater than x, and z is 4 greater than x.
So, we can write the following:
x = x
y = x + 2
z = x + 4

This means that (x + z)/y = (x + x + 4)/(x + 2)
= (2x + 4)/(x + 2)
= 2
Aha, so (x + z)/y will ALWAYS equal 2 (an integer)
So, statement II is true

III. xz is even
Since x and z are both ODD, xz = (ODD)(ODD) = ODD
So, statement III is NOT true

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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by Brent@GMATPrepNow » Wed Aug 20, 2014 7:30 am

prachi18oct wrote:If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

I. x + y is even
II. (x + z)/y is an integer
III. xz is even

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

I eliminated option D as I considered -1,1,3 as one of the possibilities and hence -1+1 = 0 which is neutral.Please suggest if we should consider this possibility.

Your mistake is highlighted above.

I believe you are confusing even/odd with positive/negative.
0 is neither positive nor negative.
But 0 is definitely even.

A number is "even" if that number can be written as the product of 2 and some integer.
Since we can write 0 as (2)(0), 0 is even.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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prachi18oct: Master | Next Rank: 500 Posts; Posts: 269; Joined: Sun Apr 27, 2014 10:33 pmFollowed by: 5 members; Thanked: 8 times

by prachi18oct » Wed Aug 20, 2014 9:15 am

Yeah, my bad!! I got confused between even & positive/negative.
Thanks Brent!!

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by [email protected] » Wed Aug 20, 2014 11:39 am

Hi prachi18oct,

Roman Numeral questions are sometimes "built" to take longer to solve than typical Quant questions. However, they almost always involve Number Properties, which can be used to save time (if you know your Number Property rules).

From the onset of the question, we're told that X, Y and Z are CONSECUTIVE ODD INTEGERS. That type of vocabulary is a HUGE clue to think about Number Properties (re "patterns" in numbers). As Brent has shown in his explanation, you can quickly prove that Roman Numeral I is true and Roman Numeral III is false.

Be on the lookout for these patterns. Number Properties tend to show up all over the Quant section (especially in DS); knowing them will give you a great way to save time and pick up points.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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j_shreyans: Legendary Member; Posts: 510; Joined: Thu Aug 07, 2014 2:24 amFollowed by: 5 members; Thanked: 3 times

by j_shreyans » Wed Aug 20, 2014 10:55 pm

Hi Brent ,

Can you pls explain why did you take y=x+2 if x=x it must be consecutive. I am bit confused in this.

Thanks

Shreyans

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Wed Aug 20, 2014 11:10 pm

j_shreyans wrote:Hi Brent ,

Can you pls explain why did you take y=x+2 if x=x it must be consecutive. I am bit confused in this.

Thanks

Shreyans

Not to step on Brent's toes, but he did this because the numbers are consecutive ODD integers, which must differ by 2 (e.g. 1 and 3 or 3 and 5).

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by Brent@GMATPrepNow » Thu Aug 21, 2014 6:19 am

j_shreyans wrote:Hi Brent ,

Can you pls explain why did you take y=x+2 if x=x it must be consecutive. I am bit confused in this.

Thanks

Shreyans

Let's examine some consecutive ODD integers: 7, 9, 11
Notice that each integer is 2 greater than the one before it.
So, for example, 9 is 2 greater than 7, and 11 is 4 greater than 7
We can write:
7 = 7
9 = 7 + 2
11 = 7 + 4

Likewise, if x, y and z are consecutive ODD integers, then:
x = x
y = x + 2
z = x + 4

Once we know this, we can take (x + z)/y and replace y with x + 2 and replace z with x + 4 to get (x + x + 4)/(x + 2), which simplifies to (2x + 4)/(x + 2), which simplifies to 2

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: consecutive odd integers

by Scott@TargetTestPrep » Fri Feb 14, 2020 1:11 pm

prachi18oct wrote: ↑
Wed Aug 20, 2014 6:41 am
If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

x + y is even
is an integer
xz is even

I only
II only
III only
I and II only
I, II, and III

I eliminated option D as I considered -1,1,3 as one of the possibilities and hence -1+1 = 0 which is neutral.Please suggest if we should consider this possibility.

(Note that 0 is an even integer, so D can not be eliminated based on your statement.)

Since x, y, and z are consecutive odd integers, x + y = odd + odd = even, and xz = (odd)(odd) = odd. So statement I is true, and statement III is false.

Since x = y - 2 and z = y + 2, then (x + z)/y = (y - 2 + y + 2)/y = 2y/y = 2 is an integer. So statement II is true also.

Answer: D

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