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

 It is currently 18 Sep 2018, 20:43

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 x is a positive integer, x^(1/2) an integer?

Author Message
TAGS:

Hide Tags

Study Buddy Forum Moderator
Joined: 04 Sep 2016
Posts: 1196
Location: India
WE: Engineering (Other)
If x is a positive integer, x^(1/2) an integer?  [#permalink]

Show Tags

Updated on: 25 Nov 2017, 00:13
00:00

Difficulty:

35% (medium)

Question Stats:

71% (01:00) correct 29% (01:08) wrong based on 69 sessions

HideShow timer Statistics

If x is a positive integer, is $$\sqrt{x}$$ an integer ?

(1) $$\sqrt{36x}$$ is an integer.

(2) $$\sqrt{3x+4}$$ is an integer

_________________

It's the journey that brings us happiness not the destination.

Originally posted by adkikani on 24 Nov 2017, 16:40.
Last edited by Bunuel on 25 Nov 2017, 00:13, edited 2 times in total.
Renamed the topic, edited the question and edited the tags.
Math Expert
Joined: 02 Aug 2009
Posts: 6786
Re: If x is a positive integer, x^(1/2) an integer?  [#permalink]

Show Tags

24 Nov 2017, 19:42
If x is a positive integer, is $$\sqrt{x}$$ an integer

1. $$\sqrt{36x}$$ is an integer.

2. $$\sqrt{3x+4}$$ is an integer

is $$\sqrt{x}$$ an integer?

1. $$\sqrt{36x}$$ is an integer.
$$\sqrt{36x}=\sqrt{6^2x}=6\sqrt{x}$$
Now if $$6\sqrt{x}$$ is a n integer and x is a positive integer, $$\sqrt{x}$$ has to be an integer
sufficient

2. $$\sqrt{3x+4}$$ is an integer
there can be various possiblities....
x = 4, $$\sqrt{3x+4} = \sqrt{3*4+4}\sqrt{16}=4$$
here $$\sqrt{x} =2$$ is an integer ...yes
x = 7, $$\sqrt{3x+4} = \sqrt{3*7+4}\sqrt{25}=5$$
here $$\sqrt{5} =2$$ is not an integer ...
insuff

A
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Study Buddy Forum Moderator
Joined: 04 Sep 2016
Posts: 1196
Location: India
WE: Engineering (Other)
Re: If x is a positive integer, x^(1/2) an integer?  [#permalink]

Show Tags

24 Nov 2017, 23:20
chetan2u niks18 Bunuel

Can any of rules be applied universally?

(1) $$\sqrt{ab}$$ = $$\sqrt{a}$$ * $$\sqrt{b}$$

is it necessary for a and b to be distinct?

(2) $$\sqrt{a+b}$$ = $$\sqrt{a}$$ + $$\sqrt{b}$$
_________________

It's the journey that brings us happiness not the destination.

Math Expert
Joined: 02 Sep 2009
Posts: 49208
Re: If x is a positive integer, x^(1/2) an integer?  [#permalink]

Show Tags

25 Nov 2017, 00:44
1
If x is a positive integer, is $$\sqrt{x}$$ an integer?

The square root of any positive integer is either an integer or an irrational number. So, $$\sqrt{integer}$$ cannot be a fraction, for example it cannot equal to 1/2, 3/7, 19/2, ... It MUST be an integer (0, 1, 2, 3, ...) or irrational number (for example $$\sqrt{2}$$, $$\sqrt{3}$$, $$\sqrt{17}$$, ...). So, as given that $$x$$ is a positive integer then $$\sqrt{x}$$ is either an integer itself or an irrational number.

(1) $$\sqrt{36x}$$ is an integer:

$$\sqrt{36x} = 6\sqrt{x}$$. As discussed above, $$\sqrt{x}$$ cannot be a fraction, say 1/6 and integer*irrational cannot be an integer. Therefore, for this statement to be true, $$\sqrt{x}$$ must be an integer. Sufficient.

(2) $$\sqrt{3x+4}$$ is an integer. If x = 4, then the answer is YES but if x = 7, then the answer is NO. Not sufficient.

Similar questions:
https://gmatclub.com/forum/if-n-p-q-p-a ... 01475.html
https://gmatclub.com/forum/if-d-is-a-po ... 09384.html
https://gmatclub.com/forum/if-x-is-a-po ... 65976.html
https://gmatclub.com/forum/if-d-is-a-po ... 67950.html
https://gmatclub.com/forum/if-z-is-a-po ... 01464.html
https://gmatclub.com/forum/if-y-is-a-po ... 08287.html
https://gmatclub.com/forum/if-sqrt-4a-i ... 06886.html
https://gmatclub.com/forum/what-is-the- ... 07195.html
https://gmatclub.com/forum/is-s-an-odd- ... 06562.html

Hope it helps.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 49208
Re: If x is a positive integer, x^(1/2) an integer?  [#permalink]

Show Tags

25 Nov 2017, 00:47
1
chetan2u niks18 Bunuel

Can any of rules be applied universally?

(1) $$\sqrt{ab}$$ = $$\sqrt{a}$$ * $$\sqrt{b}$$

is it necessary for a and b to be distinct?

(2) $$\sqrt{a+b}$$ = $$\sqrt{a}$$ + $$\sqrt{b}$$

(1) Always true: $$\sqrt{ab}$$ = $$\sqrt{a}$$ * $$\sqrt{b}$$

(2) Generally, $$\sqrt{a+b}$$ \neq $$\sqrt{a}$$ + $$\sqrt{b}$$. Does $$\sqrt{1+1}$$=$$\sqrt{1}$$ + $$\sqrt{1}$$?

8. Exponents and Roots of Numbers

Check below for more:
ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
_________________
Math Expert
Joined: 02 Aug 2009
Posts: 6786
Re: If x is a positive integer, x^(1/2) an integer?  [#permalink]

Show Tags

25 Nov 2017, 02:38
chetan2u niks18 Bunuel

Can any of rules be applied universally?

(1) $$\sqrt{ab}$$ = $$\sqrt{a}$$ * $$\sqrt{b}$$

is it necessary for a and b to be distinct?

(2) $$\sqrt{a+b}$$ = $$\sqrt{a}$$ + $$\sqrt{b}$$

Hi...

Just add on to the second point..
√a+√b will generally be NOT equal to √(a+b)..
Only exception is when either or both of them are 0..
An algebraic way to look at it...

$$√a+√b=√(a+b)$$
Square both sides..
a+b+2√(ab)=a+b........
2√(ab)=0
Only possible When either of a or b or BOTH of a and b are 0..
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Re: If x is a positive integer, x^(1/2) an integer? &nbs [#permalink] 25 Nov 2017, 02:38
Display posts from previous: Sort by

Events & Promotions

 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®.