GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 03 Jun 2020, 18:44

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

A positive integer is called "square-Free" if it has no fact

Author Message
TAGS:

Hide Tags

Director
Joined: 29 Nov 2012
Posts: 673
A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

09 Sep 2013, 23:25
3
20
00:00

Difficulty:

25% (medium)

Question Stats:

72% (01:22) correct 28% (01:31) wrong based on 519 sessions

HideShow timer Statistics

A positive integer is called "square-Free" if it has no factor that is the square of an integer greater than 1. If n is an even square-free integer, which of the following must also be square free?

A. $$\frac{n}{2}$$
B. $$2n$$
C. $$n + 2$$
D. $$n^2$$
E. none of the above

What are examples of square-free numbers?
Math Expert
Joined: 02 Sep 2009
Posts: 64224
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

09 Sep 2013, 23:44
2
fozzzy wrote:
A positive integer is called "square-Free" if it has no factor that is the square of an integer greater than 1. If n is an even square-free integer, which of the following must also be square free?

A. $$\frac{n}{2}$$
B. $$2n$$
C. $$n + 2$$
D. $$n^2$$
E. none of the above

What are examples of square-free numbers?

Per definition a square-free integer has primes in power of 1. For example 5^3 is NOT square-free because it's a multiple of 5^2.

We are told that n=2k, since n itself is square-free, then k also must be square-free --> n/2=k.

Or just pick some number for n. Say n=2, then among A, B, C, and D, only A (n/2=1) will be square free.

Hope it's clear.
_________________
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

25 Sep 2013, 12:26
2
fozzzy wrote:
A positive integer is called "square-free" if it has no factor that is the square of an integer greater than 1. If n is an even square-free integer, which of the following must also be square free?

A. $$\frac{n}{2}$$
B. $$2n$$
C. $$n + 2$$
D. $$n^2$$
E. none of the above

What are examples of square-free numbers?

kanusha wrote:
sir, pls explain this problem

Dear Kanusha,
I'm happy to respond to your p.m.

First of all, let's think about this --- the perfect squares greater than 1 are 4, 9, 16, 25, 36, 49, etc. Any numbers divisible by these would not be square-free, and any number that doesn't not happen to be a multiple of these would be square-free. Notice, in particular, all prime numbers are automatically square-free. The first few square-free numbers would be

1, 2, 3, 5, 6, 7, 10, 11, 13, 14 ,15, 17, 19, 21, 22, 23, 26, 29, 30, ....

The problem tells us that n is an even square-free numbers, so candidate could include 2, 6, 10, 14, 22, 26, 30 --- all of these are even numbers that are NOT divisible by 4, so they are the odd multiples of 2, and not even all of those, because some (such as 18) are divisible by other squares (18 is divisible by 9).

(A) n/2 --- using our example even square-free numbers above, this leads to 1, 3, 5, 7, 11, 13, 15 --- all square free --- hmmmm, this is promising
(B) Any even number time 2 is divisible by 4 --- this always leads to a number divisible by a square, i.e. NOT square-free. This is incorrect.
(C) Any odd multiple of 2, plus two, will equal an even multiple of two --- i.e. a multiple of 4, a number that is NOT square-free. This is incorrect.
(D) n^2 would be even times even, which is always divisible by four, i.e. NOT square-free. This is incorrect.
(E) hmmm.

It's very easy to eliminate (B) & (C) & (D). In order to choose (A) over (E), we have to be sure that (A) works, not just for some examples that we pick, but for all possible even square-free numbers. Here's an argument, involving factors.

Suppose n is an even square-free number. As we have seen, it must be an odd multiple of two. This means n = 2*q, where q is some odd number. Because n has no factors that are squares, this necessarily means that q has no factors that are squares. (If P = R*Q, then any factor of Q or R must be a factor of P.) Now, n/2 = q, and q has no square factors, so if n is square-free, then n/2 = q must be square-free. That's why (A) must be the correct answer.

Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Director
Joined: 29 Nov 2012
Posts: 673
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

26 Sep 2013, 00:29
These definition questions tend to be tricky. I usually get thrown off by such questions... do you have any questions for practice?
Math Expert
Joined: 02 Sep 2009
Posts: 64224
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

26 Sep 2013, 01:32
1
3
Math Expert
Joined: 02 Sep 2009
Posts: 64224
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

30 Sep 2013, 04:27
1
Director
Joined: 03 Aug 2012
Posts: 648
Concentration: General Management, General Management
GMAT 1: 630 Q47 V29
GMAT 2: 680 Q50 V32
GPA: 3.7
WE: Information Technology (Investment Banking)
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

30 Sep 2013, 10:46
1
1
Will not pop into too much:

We will do it by hit and trial:

Since 'n' is an even-square free integers.

Then check values of 'n'

n=2 ..... Can be square free
n=4 ... Cannot be since 4 is a factor of 4
n=6.... Can be square free since 6=3*2*1
n=8... Cannot be

Try n=2,6 in answer options and hence (A) it is
Non-Human User
Joined: 09 Sep 2013
Posts: 15059
Re: A positive integer is called "square-Free" if it has no fact  [#permalink]

Show Tags

26 Apr 2020, 02:54
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: A positive integer is called "square-Free" if it has no fact   [#permalink] 26 Apr 2020, 02:54