NEW HERE? LEARN MORE
closeclose
It is currently 21 Sep 2023, 21:10
Register
GMAT Club Tests
Decision Tracker
My Rewards
New posts
New comers' posts

Close

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
Your Progress

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.

Go to My Error Log Learn more
Close

Request Expert Reply

Confirm Cancel

If x, y and z are positive integers, is it true that x^2 is divisible

5 posts Go to First Unread Post
Print view
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
SORT BY:
Date
Kudos
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 89492
CAT Tests Forum Quiz Mode
If x, y and z are positive integers, is it true that x^2 is divisible [#permalink] New post  29 May 2023, 01:30
5
Bookmarks
Expert Reply
Quiz Dashboard Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
00:00
A
B
C
D
E

Difficulty:

85% (hard)

Question Stats:

43% (01:58) correct 57% (01:43) wrong based on 54 sessions

Hide Show timer Statistics

If x, y and z are positive integers, is it true that x^2 is divisible by 16?

(1) 5x = 4y
(2) 3x^2 = 8z



ShowHide Answer
Official Answer
Official Answer and Stats are available only to registered users. Register/Login.

_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
Quiz Dashboard Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
User avatar
L
zhanbo
SVP
SVP
Joined: 27 Feb 2017
Posts: 1514
Location: United States (WA)
GMAT 1: 760 Q50 V42
GMAT 2: 760 Q50 V42
GRE 1: Q169 V168

GRE 2: Q170 V170
Re: If x, y and z are positive integers, is it true that x^2 is divisible [#permalink] New post  29 May 2023, 05:04
2
Kudos
1
Bookmarks
(1) Only
Because 5 only contain a factor of 5, x must contain a factor of 4.
Therefore, x^2 is divisible by 16.
Sufficient

(2) Only
x^2 must contain a factor of 8 = 2^3.
Because x and z are both integers, x^2 must contain a factor of 16 = 2^4.
Sufficient.

The answer is (D).
User avatar
D
desertEagle
Director
Director
Joined: 14 Jun 2014
Posts: 587
Re: If x, y and z are positive integers, is it true that x^2 is divisible [#permalink] New post  01 Jun 2023, 21:28
If x, y and z are positive integers, is it true that x^2 is divisible by 16?

(1) 5x = 4y
x = 4/5 y
x is integer (given)
so x will be a multiple of 4
so x2 will be divisible by 16

Sufficient

(2) 3x^2 = 8z

x^2 = 8/3 z

x is integer so it will have primes with even powers .. minimum power of 2 can be 4.. so divisible by 16

Sufficient

Ans D
User avatar
G
SaquibHGMATWhiz
GMATWhiz Representative
Joined: 23 May 2022
Posts: 629
Location: India
GMAT 1: 760 Q51 V40
Re: If x, y and z are positive integers, is it true that x^2 is divisible [#permalink] New post  03 Jun 2023, 01:10
Expert Reply
Bunuel wrote:
If x, y and z are positive integers, is it true that x^2 is divisible by 16?
(1) 5x = 4y
(2) 3x^2 = 8z

Solution:
Pre Analysis:
  • x, y and z are positive integers
  • We are asked if \(x^2\) is divisible by 16 or not

Statement 1: \(5x=4y\)
  • According to this statement, \(x=\frac{4}{5}y\)
    \(⇒x^2=\frac{16}{25}y^2\)
    \(⇒x^2=16\times \frac{y^2}{25}\)
  • \(\frac{y^2}{25}=k\) has to be positive integer because \(x\) is a positive integer
  • Which means \(x^2=16\times k\) and \(x^2\) is divisible by 16
  • Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: \(3x^2 = 8z\)
  • According to this statement, \(x^2=\frac{8z}{3}=\frac{2^3\times z}{3}\)
  • The minimum values of z is \(2\times 3\) which makes \(x^2\) divisible by 16
  • Thus, statement 2 alone is also sufficient


Hence the right answer is Option D
_________________


Register for Free GMAT Live Sessions

Every week we conduct 3 FREE live sessions on various topics in Quant, Verbal, Strategy and Admission Consulting. You can register for the webinars - HERE

GMATWhiz FREE Trial: Prepare from an application driven course that serves real-time improvement modules along with a well-defined adaptive study plan. Start a free trial to experience it yourself and get access to 25 videos and 300 GMAT styled questions.

Score Improvement Strategy: How to score Q50+ on GMAT | 5 steps to Improve your Verbal score
Signature Read More
User avatar
P
mysterymanrog
CR Forum Moderator
Joined: 25 Jan 2022
Posts: 663
Location: Italy
Schools: HEC
GPA: 3.9
Forum Quiz Mode
Re: If x, y and z are positive integers, is it true that x^2 is divisible [#permalink] New post  04 Jun 2023, 07:59
1)
\(5x=4y\)
\(x=\frac{4y}{5}\)
\(x^2=\frac{16y^2}{25}\)

Because X is integer, we know that \(\frac{16y^2}{25}\) is an integer as well.
This implies that \(y=5*k*n*...*p\)
Y has at least 1 5 and other possible factors.
implying that
\(x^2=16*k*n*...*p\)
Which is clearly divisible by 16.
Suff.

2)
\(3x^2=8z\)
\(x^2=\frac{8z}{3}\)
\(x^2=\frac{2^3z}{3}\)
Recall that X is an integer, so x^2 is also an integer. This means that the fraction must be a perfect square.
That implies that:
\(z=2*3*k*n*...*p\)
Which in turn implies that:
\(x^2=2^4*k*n*...*p\)

Which is obviously divisible by 16.

Suff.

D
GMAT Club Bot
gmatclubot
Re: If x, y and z are positive integers, is it true that x^2 is divisible [#permalink]
04 Jun 2023, 07:59
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
Moderator:
Bunuel
Math Expert
89492 posts

cron

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions