If x, y, and z are positive integers, is x-y odd ?
(1) x = z^2
(2) y = (z - 1)^2
Thanks.
Positive/Negative: If x, y, and z are positive integers, is
This topic has expert replies
-
- Senior | Next Rank: 100 Posts
- Posts: 93
- Joined: Thu Apr 10, 2008 1:42 pm
- Location: Chicago
- Thanked: 20 times
Interesting question II.II wrote:If x, y, and z are positive integeres, is x-y odd ?
(1) x = z^2
(2) y = (z - 1)^2
Thanks.
This is the very essence of Data Sufficiency. Within 10-20 seconds you should be able to eliminte A / B and D as possible answers.
Why?
Since the relationship being tested requires us to look for the outcome of X - Y, it becomes obvious that Statement 1 and Statement 2 by themselves are not going to pull it off (they only have X and Z, and Y and Z respectively.
If on test-da your time is running out (or even if it is not), something like this is really going to help you.
Now to decide between C and E:
Together:
X - Y = Z^2 - (Z-1)^2.................(This is the classic difference of squares rule)
Therfore, X- Y = ( Z - (Z-1)) ( Z+(Z+1))
which is then = (Z - Z +1) (2Z+1)
which simplifies as = (1) (2Z +1) = 2Z +1
The odd number kingdom is defined as 2(integer) + 1 OR 2(Integer) - 1.
Since Z is an integer, 2Z+1 MUST be odd. This answers the question with an ALWAYS YES, and hence sufficiency.
C is the answer.
For love, not money.
-
- Senior | Next Rank: 100 Posts
- Posts: 39
- Joined: Sun Apr 13, 2008 12:17 am
-
- Legendary Member
- Posts: 518
- Joined: Tue May 12, 2015 8:25 pm
- Thanked: 10 times
GMAT/MBA Expert
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
Here's an algebraic approach:If x, y, and z are positive integers, is x-y odd?
1) x = z²
2) y = (z-1)²
Target question: Is x-y odd?
Given: x, y, and z are positive integers
Statement 1: x = z²
There's no information about y, so there's no way to determine whether or not x-y is odd.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: y = (z-1)²
There's no information about x, so there's no way to determine whether or not x-y is odd.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1: x = z²
Statement 2: y = (z-1)²
Subtract equations to get: x-y = z² - (z-1)²
Expand to get: x-y = z² - [z² - 2z + 1]
Simplify to get: x-y = 2z - 1
Since z is a positive integer, we know that 2z is EVEN, which means 2z-1 is ODD.
If 2z-1 is ODD, we can conclude that x-y is definitely ODD
Since we can answer the target question with certainty, the combined statements are SUFFICIENT
Answer = C
Cheers,
Brent
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
Clearly, neither statement alone is sufficient.II wrote:If x, y, and z are positive integers, is x-y odd ?
(1) x = z^2
(2) y = (z - 1)^2
Statements combined:
Since x and y are both positive integers in terms of positive integer z -- and neither statement involves division -- we can determine whether x-y must be ODD by testing two cases:
z = EVEN and z = ODD.
Case 1: z=2
Here, x = 2² = 4 and y = (2-1)² = 1.
In this case, x-y = 4-1 = 3, which is ODD.
Case 2: z=3
Here, x = 3² = 9 and y = (3-1)² = 4.
In this case, x-y = 9-4 = 5, which is ODD.
Since x-y is ODD in both cases, the answer to the question stem is YES.
Thus, the two statements combined are SUFFICIENT.
The correct answer is C.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
-
- Legendary Member
- Posts: 518
- Joined: Tue May 12, 2015 8:25 pm
- Thanked: 10 times
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
Another approach:
z² - (z-1)² =
z² - (z² - 2z + 1) =
2z - 1
For any integer z, 2z is even, so 2z - 1 is odd. We're set!
z² - (z-1)² =
z² - (z² - 2z + 1) =
2z - 1
For any integer z, 2z is even, so 2z - 1 is odd. We're set!