Events & Promotions
|
|
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
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Nov 1912:30 PM EST
-01:30 PM EST
Learn how Keshav, a Chartered Accountant, scored an impressive 705 on GMAT in just 30 days with GMATWhiz's expert guidance. In this video, he shares preparation tips and strategies that worked for him, including the mock, time management, and more
Nov 2001:30 PM EST
-02:30 PM IST
Learn how Kamakshi achieved a GMAT 675 with an impressive 96th %ile in Data Insights. Discover the unique methods and exam strategies that helped her excel in DI along with other sections for a balanced and high score.
Nov 2211:00 AM IST
-01:00 PM IST
Do RC/MSR passages scare you? e-GMAT is conducting a masterclass to help you learn – Learn effective reading strategies Tackle difficult RC & MSR with confidence Excel in timed test environment- Nov 23
11:00 AM IST
-01:00 PM IST
Attend this free GMAT Algebra Webinar and learn how to master the most challenging Inequalities and Absolute Value problems with ease. 
Nov 2407:00 PM PST
-08:00 PM PST
Full-length FE mock with insightful analytics, weakness diagnosis, and video explanations!
Nov 2510:00 AM EST
-11:00 AM EST
Prefer video-based learning? The Target Test Prep OnDemand course is a one-of-a-kind video masterclass featuring 400 hours of lecture-style teaching by Scott Woodbury-Stewart, founder of Target Test Prep and one of the most accomplished GMAT instructors.
29 Oct 2020, 11:50
Kudos
Bookmarks
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
47% (03:01) correct
53%
(03:12)
wrong
based on 192
sessions
History
Date
Time
Result
Not Attempted Yet
If two distinct integers x and y are randomly selected from the integers between 1 and 10, inclusive, what is the probability that \(\frac{x^2-y^2}{4}\) is a positive odd integer?
(A) 1/12
(B) 1/14
(C) 1/15
(D) 1/8
(E) 2/15
(A) 1/12
(B) 1/14
(C) 1/15
(D) 1/8
(E) 2/15
29 Oct 2020, 16:26
Kudos
Bookmarks
Sajjad1994
\(\frac{x^2-y^2}{4} = \frac{(x + y)(x - y) }{ 4}\).
We need (x + y)(x - y) to be divisible by 4 first of all, in order to do that we need both x and y to be both even or both odd. 1 odd 1 even will result in Odd*Odd.
However, if we have both x and y odd, one of x + y or x - y will be a multiple of 4 guaranteed. This is because while they are both even, their difference is 2y which is not a multiple of 4. If x - y is not a multiple of 4 for example, adding 2y will make x + y a multiple of 4 and vice versa. Then we have a multiple of 4 times an even number. After dividing by 4 the result will be even, thus we cannot take any x odd y odd options.
Therefore the only possible options come from x even and y even.
Next, in order to have an odd result after dividing by 4, we need to make sure (x + y) is not a multiple of 4, this guarantees (x - y) is also not a multiple of 4 because 2y is a multiple of 4 (similar logic with above).
Thus picking out even options while avoiding x+y being a multiple of 4, and x > y to guarantee a positive result, we have (4, 2), (6, 4), (8, 2), (8, 6), (10, 4), (10, 8).
There are 10 * 9 random selections since once you choose one of the 10 options for x there are only 9 options left for y. The answer should be 6/90 = 1/15
Ans: C
General Discussion
05 Nov 2020, 11:21
Kudos
Bookmarks
Sajjad1994
Solution:
In order for (x^2 - y^2)/4 to be an integer, two integers of the same parity (i.e., both odd or both even) must be selected. To show this, we can let y = x - 2n where n is a positive integer no more than 4 (notice that x and x - 2n have the same parity). Therefore, we have:
x^2 - y^2 = x^2 - (x - 2n)^2 = x^2 - (x^2 - 4nx + 4n^2) = 4nx - 4n^2 = 4n(x - n)
Notice 4n(x - n) is a multiple of 4 and therefore, when it’s divided by 4, the result, n(x - n), is an integer. However, in order for n(x - n) to be a positive odd integer integer, both n and x - n must be odd, and x - n must be positive. Since n is no more than 4, n could be only 1 or 3.
If n = 1, then x = 2, 4, 6, 8, or 10 so that x - n is odd and positive. However, x can’t be 2; otherwise, y = x - 2n would be 0. In other words, we have (x, y) = (4, 2), (6, 4), (8, 6), or (10, 8).
If n = 3,then x = 4, 6, or 8 so that x - n is odd and positive. However, x can’t be 4 or 6; otherwise, y = x - 2n would be negative or 0. In other words, we have only (x, y) = (8, 2) or (10, 4).
We see that there are only 6 ordered pairs of (x, y) such that (x^2 - y^2)/4 is a positive odd integer. Since there are 10P2 = 10 x 9 = 90 ways to choose 2 distinct integers from 1 to 10, inclusive, the probability that (x^2 - y^2)/4 is a positive odd integer is 6/90 = 1/15.
Answer: C
