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

 It is currently 20 Oct 2019, 19:38

### 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

# What is the smallest positive integer n for which √n*7! is an integer?

Author Message
TAGS:

### Hide Tags

Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1348
Location: Malaysia
What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

21 Feb 2017, 23:40
1
2
00:00

Difficulty:

5% (low)

Question Stats:

85% (01:16) correct 15% (01:32) wrong based on 162 sessions

### HideShow timer Statistics

What is the smallest positive integer n for which $$\sqrt{n*7!}$$ is an integer?

A. 14
B. 35
C. 70
D. 105
E. 210

_________________
"Be challenged at EVERY MOMENT."

“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”

"Each stage of the journey is crucial to attaining new heights of knowledge."

Manager
Joined: 05 Apr 2014
Posts: 136
Location: India
Schools: ISB '19, Fox"19
GMAT 1: 660 Q48 V33
GPA: 3
Re: What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

22 Feb 2017, 00:07
B?
Coz root (n*7!)
For this to b integer take pairs which can come out of root..we wl b left wd 5 and 7 ...which need another 5 n 7 to form sqr and come out

Sent from my SM-G600FY using GMAT Club Forum mobile app
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4015
Re: What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

22 Feb 2017, 09:05
2
Top Contributor
1
AustinKL wrote:
What is the smallest positive integer n for which $$\sqrt{n * 7!}$$ is an integer?

A. 14
B. 35
C. 70
D. 105
E. 210

IMPORTANT CONCEPTS:
#1) In order for √K to be an integer, K must be a perfect square (e.g., 1, 4, 9, 16, 25, 36, 49, etc)

#2) The prime factorization of a perfect square will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even numbers allow us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2X2X2X2X2X2X3X3 = (2X2X2X3)(2X2X2X3) = (2X2X2X3)²

---ONTO THE QUESTION!!----------------------------------------

For the original question, we must recognize that n * 7! must be a perfect square.
n * 7! = (n)(7)(6)(5)(4)(3)(2)(1)
= (n)(7)(3)(2)(5)(2)(2)(3)(2)
= (n)(2)(2)(2)(2)(3)(3)(5)(7)

As we can see, we already have an EVEN number of 2's and 3's.
So, to get an even number of all of the primes (and thus create a perfect square), we need to add a 5 and a 7 to the prime factorization of n * 7!
So, let n = (5)(7) = 35
This means n * 7! = (2)(2)(2)(2)(3)(3)(5)(5)(7)(7), in which case it is a perfect square AND $$\sqrt{n * 7!}$$ is an integer

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4774
Location: India
GPA: 3.5
Re: What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

22 Feb 2017, 10:24
AustinKL wrote:
What is the smallest positive integer n for which $$\sqrt{n*7!}$$ is an integer?

A. 14
B. 35
C. 70
D. 105
E. 210

$$= \sqrt{n*7*6*5*4*3*2}$$

$$= \sqrt{n*7*2*3*5*2*2*3*2}$$

$$= \sqrt{n*7*5*3^2*2^4}$$

$$= 2^2*3\sqrt{n*7*5}$$

Thus, n must be 35, answer must be (B) 35
_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Director
Joined: 12 Nov 2016
Posts: 699
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37
GRE 1: Q157 V158
GPA: 2.66
Re: What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

15 Apr 2017, 17:27
ziyuen wrote:
What is the smallest positive integer n for which $$\sqrt{n*7!}$$ is an integer?

A. 14
B. 35
C. 70
D. 105
E. 210

We can solve this question if we break down the components of 7!

[square_root] 7 x 6 x 5 x 4 x 3 x 2 x 1
[square_root] 7 x (3 x 2) x 5 x (2 x 2) x 3 x 2 x 1
[square_root] 5 x 7 x 3^ 2 x 2^4 x n

However, the exponents of the factors a perfect square MUST be multiples of 2- therefore the factors of N must necessarily contain factors of 5 and 7 so as to make the powers of abundance or powers of 5'2 and 7's in the number squared multiples of 2.

If we analyze 14-
Factors: 1, 2, 7 , 14 - cannot be the answer

If we analyze 35-
Factors: 1, 5, 7 , 35

So if N is 35 then

[square_root] 5^2 x 7^2 x 3^ 2 x 2^4

Now, because the exponents are multiples of two this constitutes a perfect square.
Director
Joined: 12 Nov 2016
Posts: 699
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37
GRE 1: Q157 V158
GPA: 2.66
Re: What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

19 Jun 2017, 08:51
hazelnut wrote:
What is the smallest positive integer n for which $$\sqrt{n*7!}$$ is an integer?

A. 14
B. 35
C. 70
D. 105
E. 210

We can simply break this apart and apply a basic property of square roots

\sqrt{7 x 6 x 5 x 4 x 3 x 2 x 1}

\sqrt{7 x 6 x 5 x 4 x 6}

\sqrt{7 x 6^2 x 5 x 4^2} - now a basic property of any perfect square root is that the number of odd integers must be even for example 36 = \sqrt{3^2 x 2^2}

If we apply 35 which is 7 and 5 then our answer is

\sqrt{7^2 x 6^2 x 5^2 x 4^2}

Thus
"B"
Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4774
Location: India
GPA: 3.5
Re: What is the smallest positive integer n for which √n*7! is an integer?  [#permalink]

### Show Tags

19 Jun 2017, 09:59
hazelnut wrote:
What is the smallest positive integer n for which $$\sqrt{n*7!}$$ is an integer?

A. 14
B. 35
C. 70
D. 105
E. 210

$$7! = 7*6*5*4*3*2$$

Or, $$7! = 7*(2*3)*5*2^2*3*2$$

Or, $$7! = 7*5*3^2*2^4$$

Thus, the minimum value of $$\sqrt{n*7!}$$ must be 35, answer will be (B)

_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Re: What is the smallest positive integer n for which √n*7! is an integer?   [#permalink] 19 Jun 2017, 09:59
Display posts from previous: Sort by