What is the smallest positive integer

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 am; Followed by:3 members

What is the smallest positive integer

by VJesus12 » Thu Dec 28, 2017 4:47 am

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What is the smallest positive integer n for which $$\sqrt{n\cdot7!}\ is\ an\ integer?$$ A. 14
B. 35
C. 70
D. 105
E. 210

The OA is the option B.

How can I find the value of n? Should I try number by number? Experts, may you give me some help, please?

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Source: Beat The GMAT — Problem Solving | Thu Dec 28, 2017 4:47 am

EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pm; Thanked: 180 times; Followed by:12 members

by EconomistGMATTutor » Thu Dec 28, 2017 8:45 am

Hello Vjesus12.

Let's see your question.

You could try number by number or solve it as follows:

$$\sqrt{n\cdot7!}=\sqrt{n\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}=2\sqrt{n\cdot7\cdot(3\cdot2)\cdot5\cdot3\cdot2}$$ $$2\sqrt{n\cdot7\cdot2\cdot3\cdot5\cdot3\cdot2}=2\sqrt{n\cdot7\cdot4\cdot9\cdot5}=2\cdot2\cdot3\sqrt{n\cdot7\cdot5}=12\sqrt{n\cdot7\cdot5}.$$ Now, this last number is an interger when the prime factorizazion has an odd number of 5's and a 7's.

On the given list, the smallest number that holds this condition is 35.

So, the correct answer is [spoiler]B=35[/spoiler].

I hope this explanation may help you.

I'm available if you'd like a follow-up.

Regards.

GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8083; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Wed Sep 04, 2019 5:17 pm

VJesus12 wrote:What is the smallest positive integer n for which $$\sqrt{n\cdot7!}\ is\ an\ integer?$$ A. 14
B. 35
C. 70
D. 105
E. 210

The OA is the option B.

How can I find the value of n? Should I try number by number? Experts, may you give me some help, please?

A perfect square has a prime decomposition in which each exponent is a multiple of 2.

7! = 7 x 6 x 5 x 4 x 3 x 2 = 2^4 x 3^2 x 5 x 7, so the smallest value of n must be 5 x 7 = 35.

Answer: B

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