How many positive integers less than 30 have no

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swerve: Legendary Member; Posts: 2499; Joined: Sun Oct 29, 2017 2:04 pm; Followed by:6 members

How many positive integers less than 30 have no

by swerve » Mon Jul 09, 2018 2:47 pm

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How many positive integers less than 30 have no common prime factor with 30?

A. 5
B. 6
C. 7
D. 8
E. 9

The OA is D.

Please, can anyone explain this PS question? I tried to solve it but I can't get the correct answer. I need help. Thanks.

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Source: Beat The GMAT — Problem Solving | Mon Jul 09, 2018 2:47 pm

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pm; Followed by:5 members

by deloitte247 » Tue Jul 10, 2018 11:40 am

We are looking for positive integers less than 30 which have no common prime factors with 30.
Prime factors of 30 = 2, 3, 5
Numbers between 1 and 29 with no common prime factor = 1, 7, 11, 13, 17, 19 , 23, 29
That a total of 8 positive integers.

Answer is option D

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

by Vincen » Tue Jul 10, 2018 11:30 pm

swerve wrote:How many positive integers less than 30 have no common prime factor with 30?

A. 5
B. 6
C. 7
D. 8
E. 9

The OA is D.

Please, can anyone explain this PS question? I tried to solve it but I can't get the correct answer. I need help. Thanks.

Hi swerve.

The first step is to find the prime factors of 30. This can be done as follows: $$30=2\cdot15=2\cdot3\cdot5.$$ So, the prime factors of 30 are 2, 3 and 5.

Now, from 1 to 29, we must discard all even numbers, since 2 is a factor of all of them.

Also, we must discard all the multiples of 3 and all the multiples of 5.

Finally, we only have the numbers: 1, 7, 11, 13, 17, 19, 23 and 29.

So, there are 8 numbers that hold the condition.

The correct answer is the option D.

I hope it helps.

Regards.

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 am; Location: New York, NY; Thanked: 39 times; Followed by:22 members

by Jeff@TargetTestPrep » Sat Jul 14, 2018 6:14 pm

swerve wrote:How many positive integers less than 30 have no common prime factor with 30?

A. 5
B. 6
C. 7
D. 8
E. 9

We need to determine the number of positive integers less than 30 that are relatively prime to 30. That is, the greatest common factor of each of those numbers and 30 is 1. We can see that these numbers are:

1, 7, 11, 13, 17, 19, 23, 29

So there are 8 such numbers.

Answer: D

Jeffrey Miller
Head of GMAT Instruction
[email protected]