It is currently 20 Oct 2017, 07:57

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.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If n=s^a*t^b, where a, b, s and t are integers, is √n an integer?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Expert Post
Math Revolution GMAT Instructor
User avatar
P
Joined: 16 Aug 2015
Posts: 4142

Kudos [?]: 2895 [0], given: 0

GPA: 3.82
Premium Member CAT Tests
If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 29 May 2017, 00:00
Expert's post
4
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

29% (00:57) correct 71% (01:06) wrong based on 195 sessions

HideShow timer Statistics

If \(n=s^a t^b\), where \(a\), \(b\), \(s\) and \(t\) are integers, is \(\sqrt{n}\) an integer?

1) \(a+b\) is an even number
2) \(a\) is an even number
[Reveal] Spoiler: OA

_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Find a 10% off coupon code for GMAT Club members.
“Receive 5 Math Questions & Solutions Daily”
Unlimited Access to over 120 free video lessons - try it yourself
See our Youtube demo


Last edited by MathRevolution on 29 May 2017, 10:33, edited 2 times in total.

Kudos [?]: 2895 [0], given: 0

Senior Manager
Senior Manager
avatar
S
Joined: 22 Aug 2013
Posts: 437

Kudos [?]: 131 [0], given: 260

Location: India
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 29 May 2017, 00:22
Hi

I think there is some typo. Do you refer to 'r' or to 't'?

Kudos [?]: 131 [0], given: 260

Intern
Intern
avatar
B
Joined: 29 Mar 2017
Posts: 28

Kudos [?]: 5 [0], given: 5

GMAT 1: 750 Q50 V40
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 29 May 2017, 03:23
Can someone please explain why the answer isn't C?

My logic is as follows:

We know that all variables are integers. If n = (s^a)(t^b), then for sqrt(n) to be an integer (s^(a/2)) and (t^(b/2)) must both be integers, which means that "a" and "b" must be divisible by 2.

Statement 1: a+b is an even number. Two scenarios are possible. Either "a" and "b" are both even or both odd. If both even then N must be an integer, if both odd then N will not be an integer. Insufficient

Statement 2: "a" is an even number. This tells us nothing about "b". Insufficient.

Combining the two statements: If "a" is even, then for a+b to be even, "b" must also be even. If "a" and "b" are both even, it means that both are divisible by 2, which means that "s^a" and "t^b" both have integer roots. Sufficient.

Can someone please check my logic and let me know if I'm not seeing something? Thanks!

Kudos [?]: 5 [0], given: 5

Expert Post
1 KUDOS received
Math Revolution GMAT Instructor
User avatar
P
Joined: 16 Aug 2015
Posts: 4142

Kudos [?]: 2895 [1], given: 0

GPA: 3.82
Premium Member CAT Tests
If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 31 May 2017, 00:53
1
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
==>In the original condition, there are 5 variables (n,s,t,a,b) and 1 equation (n=s^at^b). In order to match the number of variables to the number of equations, there must be 5 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer.
By solving con 1) and con 2), s=t=2 and a=b=2 yes, but s=t=2 and a=b=-2 no, hence it is not sufficient. Therefore, the answer is E.

Answer: E
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Find a 10% off coupon code for GMAT Club members.
“Receive 5 Math Questions & Solutions Daily”
Unlimited Access to over 120 free video lessons - try it yourself
See our Youtube demo

Kudos [?]: 2895 [1], given: 0

3 KUDOS received
Intern
Intern
avatar
B
Joined: 20 Aug 2016
Posts: 8

Kudos [?]: 11 [3], given: 6

Location: India
Concentration: Strategy, International Business
WE: Operations (Computer Software)
GMAT ToolKit User
If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 15 Jun 2017, 00:56
3
This post received
KUDOS
n=s^at^b

Statement 1:

a+b=even it can happen only with below case

case 1: odd+odd = even
case 2: even+even= even

but since we are not sure what is a and b i.e even or odd so statement 1 is insufficient

Statement 2:

a is even

we don't know about b if it is even or odd.

so statement 2 is also insufficient

combining statement+ statement 2

now we know that since a is even then b is also even for a+b to be even.

BUT even if we know the values of a and b we are not aware about the values of S and T.

a = even, b= even, s=1/4 , t=1/2 then n=s^at^b is not an integer
a=even, b=even, s=2, t=3 then n=s^at^b is a integer

so the answer is E

Hope it is clear...

Kudo if you like the explanation... :-D :)

Kudos [?]: 11 [3], given: 6

VP
VP
avatar
G
Joined: 26 Mar 2013
Posts: 1260

Kudos [?]: 285 [0], given: 163

Reviews Badge CAT Tests
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 15 Jun 2017, 02:30
lovenish wrote:
n=s^at^b

Statement 1:

a+b=even it can happen only with below case

case 1: odd+odd = even
case 2: even+even= even

but since we are not sure what is a and b i.e even or odd so statement 1 is insufficient

Statement 2:

a is even

we don't know about b if it is even or odd.

so statement 2 is also insufficient

combining statement+ statement 2

now we know that since a is even then b is also even for a+b to be even.

BUT even if we know the values of a and b we are not aware about the values of S and T.

a = even, b= even, s=1/4 , t=1/2 then n=s^at^b is not an integer
a=even, b=even, s=2, t=3 then n=s^at^b is a integer

so the answer is E

Hope it is clear...

Kudo if you like the explanation... :-D :)


The highlighted is not valid as prompt stats s & t are Integers.

Kudos [?]: 285 [0], given: 163

2 KUDOS received
Intern
Intern
avatar
B
Joined: 20 Aug 2016
Posts: 8

Kudos [?]: 11 [2], given: 6

Location: India
Concentration: Strategy, International Business
WE: Operations (Computer Software)
GMAT ToolKit User
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 15 Jun 2017, 03:20
2
This post received
KUDOS
1
This post was
BOOKMARKED
n=s^at^b

Statement 1:

a+b=even it can happen only with below case

case 1: odd+odd = even
case 2: even+even= even

but since we are not sure what is a and b i.e even or odd so statement 1 is insufficient

Statement 2:

a is even

we don't know about b if it is even or odd.

so statement 2 is also insufficient

combining statement+ statement 2

now we know that since a is even then b is also even for a+b to be even.

BUT even if we know the values of a and b we are not aware about the values of S and T.

a = even, b= even, s=1/4 , t=1/2 then n=s^at^b is not an integer ----- yes this case is not possible as it is stated that s and t are integers {Thanks M02men for correcting me}.
a=even, b=even, s=2, t=3 then n=s^at^b is a integer

case :
since a and b are even integer after combining both statement 1 and statement 2.
We need to consider the negative even integers also i.e -2, -4, -6
And something raise to negative power will turn into fraction example : 2^(-1)=1/2
So in equation n=(s^a)(t^b)
If a and b are negative even integers then equation will be
n=s^(-a)t^(-b)------- n=1/(s^a)(t^b)
so now if consider some random even values for a and b like
a= -2, b=-2 , s=3, t=4 then n=3^(-2)4^(-2)-------n=1/(3^2)(4^2) now square root of n is not an integer
a=2 , b=2, s=3, t=4 then n=(3^2)(4^2)- now square root of n is a integer
so both combined together is also insufficient


so the answer is E

Hope it is clear...

Kudo if you like the explanation...

Kudos [?]: 11 [2], given: 6

Senior Manager
Senior Manager
User avatar
S
Joined: 04 Oct 2015
Posts: 394

Kudos [?]: 24 [0], given: 232

Location: Viet Nam
Concentration: Finance, Economics
GPA: 3.56
Premium Member Reviews Badge CAT Tests
If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 17 Jul 2017, 19:25
Pay attention to SPECIAL CASE in which a or b is NEGATIVE integer!!!

Some SPECIAL CASES:

x = y

x = 0, +1, -1
_________________

Do not pray for an easy life, pray for the strength to endure a difficult one - Bruce Lee

Kudos [?]: 24 [0], given: 232

Manager
Manager
avatar
B
Joined: 22 Sep 2016
Posts: 215

Kudos [?]: 24 [0], given: 42

Location: India
GMAT 1: 710 Q50 V35
GPA: 4
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 02 Aug 2017, 05:50
lovenish wrote:
n=s^at^b

Statement 1:

a+b=even it can happen only with below case

case 1: odd+odd = even
case 2: even+even= even

but since we are not sure what is a and b i.e even or odd so statement 1 is insufficient

Statement 2:

a is even

we don't know about b if it is even or odd.

so statement 2 is also insufficient

combining statement+ statement 2

now we know that since a is even then b is also even for a+b to be even.

BUT even if we know the values of a and b we are not aware about the values of S and T.

a = even, b= even, s=1/4 , t=1/2 then n=s^at^b is not an integer
a=even, b=even, s=2, t=3 then n=s^at^b is a integer

so the answer is E

Hope it is clear...

Kudo if you like the explanation... :-D :)



The question clearly mentions that a,b,s,t are INTEGERS.
The answer should be C.

chetan2u please help a bit here.
_________________

Desperately need 'KUDOS' !!

Kudos [?]: 24 [0], given: 42

Intern
Intern
avatar
B
Joined: 20 Aug 2016
Posts: 8

Kudos [?]: 11 [0], given: 6

Location: India
Concentration: Strategy, International Business
WE: Operations (Computer Software)
GMAT ToolKit User
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 02 Aug 2017, 09:11
rekhabishop wrote:
lovenish wrote:
n=s^at^b

Statement 1:

a+b=even it can happen only with below case

case 1: odd+odd = even
case 2: even+even= even

but since we are not sure what is a and b i.e even or odd so statement 1 is insufficient

Statement 2:

a is even

we don't know about b if it is even or odd.

so statement 2 is also insufficient

combining statement+ statement 2

now we know that since a is even then b is also even for a+b to be even.

BUT even if we know the values of a and b we are not aware about the values of S and T.

a = even, b= even, s=1/4 , t=1/2 then n=s^at^b is not an integer
a=even, b=even, s=2, t=3 then n=s^at^b is a integer

so the answer is E

Hope it is clear...

Kudo if you like the explanation... :-D :)



The question clearly mentions that a,b,s,t are INTEGERS.
The answer should be C.

chetan2u please help a bit here.




Hi Rekhabishop

I've done some correction in my answer to my other post in the same thread

have a look at the below cases.

case :
since a and b are even integer after combining both statement 1 and statement 2.
We need to consider the negative even integers also i.e -2, -4, -6
And something raise to negative power will turn into fraction example : 2^(-1)=1/2
So in equation n=(s^a)(t^b)
If a and b are negative even integers then equation will be
n=s^(-a)t^(-b)------- n=1/(s^a)(t^b)
so now if consider some random even values for a and b like
a= -2, b=-2 , s=3, t=4 then n=3^(-2)4^(-2)-------n=1/(3^2)(4^2) now square root of n is not an integer
a=2 , b=2, s=3, t=4 then n=(3^2)(4^2)- now square root of n is a integer
so both combined together is also insufficient


so the answer is E

Hope it is clear...

Kudo if you like the explanation...

chetan2u :-D Please put your comments on this and correct me if i am missing something :-D :) :o

Kudos [?]: 11 [0], given: 6

Manager
Manager
avatar
S
Joined: 13 Mar 2013
Posts: 178

Kudos [?]: 75 [0], given: 25

Location: United States
Concentration: Leadership, Technology
GPA: 3.5
WE: Engineering (Telecommunications)
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 12 Aug 2017, 08:15
Consider the both case when a,b is positive and a,b is negative
then apply the condition given in the equation .
Note : a, b can be even and positive
OR
a,b can be even and negative . when negative , then n will not be integer .

Hence the answer should be E
_________________

Regards ,

Kudos [?]: 75 [0], given: 25

Intern
Intern
avatar
S
Joined: 08 Mar 2016
Posts: 38

Kudos [?]: 5 [0], given: 13

CAT Tests
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 12 Aug 2017, 08:32
TheKingInTheNorth wrote:
Consider the both case when a,b is positive and a,b is negative
then apply the condition given in the equation .
Note : a, b can be even and positive
OR
a,b can be even and negative . when negative , then n will not be integer .

Hence the answer should be E



So in GMAT odd and even integers are not neccesarily always positive? Only prime numbers have to be positive. Correct?

Kudos [?]: 5 [0], given: 13

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 41891

Kudos [?]: 128980 [0], given: 12185

Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 12 Aug 2017, 09:14
SOUMYAJIT_ wrote:
TheKingInTheNorth wrote:
Consider the both case when a,b is positive and a,b is negative
then apply the condition given in the equation .
Note : a, b can be even and positive
OR
a,b can be even and negative . when negative , then n will not be integer .

Hence the answer should be E



So in GMAT odd and even integers are not neccesarily always positive? Only prime numbers have to be positive. Correct?


GMAT does not have its own math, the above is generally true.

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. So, ..., -4, -2, 0, 2, 4, ... are all even integers.

An odd number is an integer that is not evenly divisible by 2. So, ..., -3, -1, 1, 3, 5, ... are all odd integers.

A Prime number is a positive integer with exactly two distinct divisors: 1 and itself.

For more check here:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 128980 [0], given: 12185

Intern
Intern
avatar
S
Joined: 08 Mar 2016
Posts: 38

Kudos [?]: 5 [0], given: 13

CAT Tests
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 12 Aug 2017, 09:20
Bunuel wrote:
SOUMYAJIT_ wrote:
TheKingInTheNorth wrote:
Consider the both case when a,b is positive and a,b is negative
then apply the condition given in the equation .
Note : a, b can be even and positive
OR
a,b can be even and negative . when negative , then n will not be integer .

Hence the answer should be E



So in GMAT odd and even integers are not neccesarily always positive? Only prime numbers have to be positive. Correct?


GMAT does not have its own math, the above is generally true.

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. So, ..., -4, -2, 0, 2, 4, ... are all even integers.

An odd number is an integer that is not evenly divisible by 2. So, ..., -3, -1, 1, 3, 5, ... are all odd integers.

A Prime number is a positive integer with exactly two distinct divisors: 1 and itself.

For more check here:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.


Yes, thanks , makes sense !


Sent from my iPhone using GMAT Club Forum mobile app

Kudos [?]: 5 [0], given: 13

Manager
Manager
avatar
G
Joined: 14 Oct 2012
Posts: 166

Kudos [?]: 50 [0], given: 942

Premium Member Reviews Badge CAT Tests
Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 13 Aug 2017, 10:01
NOTE: we can also see this problem as IS n = perfect square =>
Thus n = (s^a)*(t^b) MUST have powers (a + b) whose sum is even and ALSO factors s and t be +ve prime numbers. Now we know that (a + b) = even such that both a and b are even, but we don’t know anything about s and t except that s and t are integers. They can be +ve or -ve or non-prime!!!
Thus E

Kudos [?]: 50 [0], given: 942

Intern
Intern
avatar
B
Joined: 25 Apr 2017
Posts: 8

Kudos [?]: 0 [0], given: 17

Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer? [#permalink]

Show Tags

New post 22 Aug 2017, 16:24
lovenish wrote:
n=s^at^b

Statement 1:

a+b=even it can happen only with below case

case 1: odd+odd = even
case 2: even+even= even

but since we are not sure what is a and b i.e even or odd so statement 1 is insufficient

Statement 2:

a is even

we don't know about b if it is even or odd.

so statement 2 is also insufficient

combining statement+ statement 2

now we know that since a is even then b is also even for a+b to be even.

BUT even if we know the values of a and b we are not aware about the values of S and T.

a = even, b= even, s=1/4 , t=1/2 then n=s^at^b is not an integer
a=even, b=even, s=2, t=3 then n=s^at^b is a integer

so the answer is E

Hope it is clear...

Kudo if you like the explanation... :-D :)


Thank you for this wonderful and clear explanation. Very wonderful. Was confused after reading the original poster's explanation. You are a true GMAT hero.

Kudos [?]: 0 [0], given: 17

Re: If n=s^a*t^b, where a, b, s and t are integers, is √n an integer?   [#permalink] 22 Aug 2017, 16:24
Display posts from previous: Sort by

If n=s^a*t^b, where a, b, s and t are integers, is √n an integer?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.