GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Jun 2019, 00:14

### 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

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

# If k is a positive integer, is k the square of an integer?

Author Message
TAGS:

### Hide Tags

VP
Joined: 21 Jul 2006
Posts: 1335
If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

Updated on: 01 Mar 2012, 01:08
1
29
00:00

Difficulty:

55% (hard)

Question Stats:

59% (01:39) correct 41% (01:34) wrong based on 778 sessions

### HideShow timer Statistics

If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4.

(2) k is divisble by exactly four different prime numbers.

Originally posted by tarek99 on 21 Nov 2007, 16:23.
Last edited by Bunuel on 01 Mar 2012, 01:08, edited 1 time in total.
Edited the question and added the OA
Math Expert
Joined: 02 Sep 2009
Posts: 55681
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

01 Mar 2012, 01:11
13
10
tarek99 wrote:
If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4.

(2) k is divisble by exactly four different prime numbers.

Responding to a pm.

If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4. If $$k=4$$ answer is YES, but if $$k=8$$ answers is NO. Not sufficient.

(2) k is divisible by exactly four different prime numbers.

We don't know the powers of these primes, so if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES ($$k$$ equals to square of some integer). Not sufficient.

(1)+(2) Again if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES. Not sufficient.

Hope it's clear.
_________________
##### General Discussion
Director
Joined: 08 Jun 2007
Posts: 531
Re: DS- is k the square of an integer?  [#permalink]

### Show Tags

21 Nov 2007, 17:43
1
tarek99 wrote:
If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4

(2) k is divisble by exactly four different prime numbers.

B for me.
1) doesnt make any point..can be 4 8 16 20..not suff

2) for a number which is divisible by 4 different prime numbers ,its square cannot be integer.
Manager
Joined: 03 Sep 2006
Posts: 224

### Show Tags

21 Nov 2007, 20:00
B for me

I. k could be 8 or could be 16 - INS
II. K could be 2*3*5*7 or 3*5*7*11 (so do others). Neither number is a perfect square, hence my ans is B
Intern
Joined: 06 Sep 2007
Posts: 8

### Show Tags

21 Nov 2007, 22:24
If k is 2^2*3^2*5^2*7^2 ie (2*3*5*7)^2

on this condition, is B right?
Director
Joined: 03 May 2007
Posts: 763
Schools: University of Chicago, Wharton School
Re: DS- is k the square of an integer?  [#permalink]

### Show Tags

22 Nov 2007, 00:07
tarek99 wrote:
If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4

(2) k is divisble by exactly four different prime numbers.

E. K could be (2x3x5x7) or (2x2x3x5x7) or (2x3x5x7)^2.
VP
Joined: 21 Jul 2006
Posts: 1335

### Show Tags

22 Nov 2007, 03:00
the OA is E. but i don't understand how. can anyone provide the detailed steps so that we can learn from this? it would be cool for all of us.

thanks
VP
Joined: 21 Jul 2006
Posts: 1335

### Show Tags

22 Nov 2007, 03:33
1
Fistail, so when the question mentions for 4 different prime numbers, we can actually have 6 prime numbers in total but could be repeated numbers or have exactly 4 prime numbers that are different. so if the question specifies that there are only 4 prime numbers and each is different, then we could answer this with C because we can never have such a number that is a multiple of 4 because we only have 1 even number, which is 2, therefore, C would say that such a combination can not be possible. correct?
Intern
Joined: 10 Jan 2012
Posts: 38
Location: United States
Concentration: Finance, Entrepreneurship
Schools: Jones '15 (M)
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

10 Apr 2012, 12:21
Bunuel wrote:

(2) k is divisible by exactly four different prime numbers.

We don't know the powers of these primes, so if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES ($$k$$ equals to square of some integer). Not sufficient.

This is what I was confused by - so even though you have multiple prime numbers as in (2^2*3*5*7)^2 this is still considered exactly four different prime numbers? Therefore stat 2 is insuff?

Or put differently if I were to ask someone which scenario has exactly four different prime numbers
1 - 2^2*3*5*7
2 - (2^2*3*5*7)^2
the answer would be that they both have four different prime numbers?

Thank you
Math Expert
Joined: 02 Sep 2009
Posts: 55681
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

10 Apr 2012, 12:28
1
destroyerofgmat wrote:
Bunuel wrote:

(2) k is divisible by exactly four different prime numbers.

We don't know the powers of these primes, so if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES ($$k$$ equals to square of some integer). Not sufficient.

This is what I was confused by - so even though you have multiple prime numbers as in (2^2*3*5*7)^2 this is still considered exactly four different prime numbers? Therefore stat 2 is insuff?

Or put differently if I were to ask someone which scenario has exactly four different prime numbers
1 - 2^2*3*5*7
2 - (2^2*3*5*7)^2
the answer would be that they both have four different prime numbers?

Thank you

Yes, both 2^2*3*5*7 and (2^2*3*5*7)^2 have 4 different primes, namely; 2, 3, 5, and 7. How else?

Consider this: 2, 4, 8, and 16 all have only one distinct prime, which is 2.

Hope it's clear.
_________________
Retired Moderator
Joined: 10 May 2010
Posts: 807
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

10 Apr 2012, 12:30
Exactly 4 different prime numbers means divisible by only 4 different prime numbers and no other factors. But again this is not possible as if you have 4 different prime factors then you can also have non prime factors. Also 1 is a factor of all the numbers.
_________________
The question is not can you rise up to iconic! The real question is will you ?
Math Expert
Joined: 02 Sep 2009
Posts: 55681
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

10 Apr 2012, 12:36
AbhiJ wrote:
Exactly 4 different prime numbers means divisible by only 4 different prime numbers and no other factors. But again this is not possible as if you have 4 different prime factors then you can also have non prime factors. Also 1 is a factor of all the numbers.

That's not correct.

For example: 6 is divisible by exactly two distinct prime factors 2 and 3, but this doesn't mean that 6 divisible by ONLY two factors each of which is a prime.
_________________
Intern
Joined: 10 Jan 2012
Posts: 38
Location: United States
Concentration: Finance, Entrepreneurship
Schools: Jones '15 (M)
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

10 Apr 2012, 12:45
1
Bunuel wrote:

Yes, both 2^2*3*5*7 and (2^2*3*5*7)^2 have 4 different primes, namely; 2, 3, 5, and 7. How else?

Consider this: 2, 4, 8, and 16 all have only one distinct prime, which is 2.

Hope it's clear.

Thanks Bunuel. Another kudos to you, fine sir.

I guess I was thinking that squaring changes TOTAL factors but not different prime factors.
Math Expert
Joined: 02 Sep 2009
Posts: 55681
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

10 Apr 2012, 12:56
destroyerofgmat wrote:
Bunuel wrote:

Yes, both 2^2*3*5*7 and (2^2*3*5*7)^2 have 4 different primes, namely; 2, 3, 5, and 7. How else?

Consider this: 2, 4, 8, and 16 all have only one distinct prime, which is 2.

Hope it's clear.

Thanks Bunuel. Another kudos to you, fine sir.

I guess I was thinking that squaring changes TOTAL factors but not different prime factors.

If $$a$$ and $$b$$ are positive integers then $$a^b$$ will have as many different prime factors as $$a$$ itself, exponentiation doesn't "produce" primes.

For example: 6, 6^2, 6^3, ..., 6^100 will all have only two distinct primes: 2 and 3. Though total # of factors will naturally be different: # of factors of 6=2*3 is 4, # of factors of 6^2=2^2*3^2 is (2+1)(2+1)=9, ...

For more on this check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
_________________
Manager
Joined: 10 Mar 2014
Posts: 186
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

26 Apr 2014, 05:16
Bunuel wrote:
tarek99 wrote:
If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4.

(2) k is divisble by exactly four different prime numbers.

Responding to a pm.

If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4. If $$k=4$$ answer is YES, but if $$k=8$$ answers is NO. Not sufficient.

(2) k is divisible by exactly four different prime numbers.

We don't know the powers of these primes, so if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES ($$k$$ equals to square of some integer). Not sufficient.

(1)+(2) Again if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES. Not sufficient.

Hope it's clear.

Hi Bunnel,

I have a doubt. following are the details

1.As in your previous post you have suggested a perfect square will always have odd number of multiples. Now we have 4 different prime numbers as factor

ex. 2^1 *3^1* 5^1*7^1 so number of factors=( 2*2*2*2) = 16 can not be perfect square we can take diff. powers such as 2,3,4 and verify

we can consider this or not?

Thanks.
Math Expert
Joined: 02 Sep 2009
Posts: 55681
If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

26 Apr 2014, 09:48
PathFinder007 wrote:
Bunuel wrote:
tarek99 wrote:
If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4.

(2) k is divisble by exactly four different prime numbers.

Responding to a pm.

If k is a positive integer, is k the square of an integer?

(1) k is divisible by 4. If $$k=4$$ answer is YES, but if $$k=8$$ answers is NO. Not sufficient.

(2) k is divisible by exactly four different prime numbers.

We don't know the powers of these primes, so if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES ($$k$$ equals to square of some integer). Not sufficient.

(1)+(2) Again if $$k=2^2*3*5*7$$ the answer is NO, but if $$k=(2^2*3*5*7)^2$$ the answer is YES. Not sufficient.

Hope it's clear.

Hi Bunnel,

I have a doubt. following are the details

1.As in your previous post you have suggested a perfect square will always have odd number of multiples. Now we have 4 different prime numbers as factor

ex. 2^1 *3^1* 5^1*7^1 so number of factors=( 2*2*2*2) = 16 can not be perfect square we can take diff. powers such as 2,3,4 and verify

we can consider this or not?

Thanks.

Firs of all a prefect square has odd number of factors, not multiples, the number of multiples is not limited for any integer.

Next, I don't quite understand how are you trying to use that in your solution. Anyway, in my post there are two examples given: one is not a prefect square ($$k=2^2*3*5*7$$) and another is ($$k=(2^2*3*5*7)^2$$).
_________________
Manager
Joined: 30 Jul 2014
Posts: 121
GPA: 3.72
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

08 Sep 2017, 05:27
I failed to interpret the "accurate" meaning of the sentence - "(2) k is divisble by exactly four different prime numbers.". I assumed that there are "only" four divisors of the number - hence marked option C - that is incorrect.
_________________
A lot needs to be learned from all of you.
Non-Human User
Joined: 09 Sep 2013
Posts: 11396
Re: If k is a positive integer, is k the square of an integer?  [#permalink]

### Show Tags

01 Dec 2018, 06:52
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If k is a positive integer, is k the square of an integer?   [#permalink] 01 Dec 2018, 06:52
Display posts from previous: Sort by