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How many positive integers less than 30 have no common prime

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How many positive integers less than 30 have no common prime [#permalink]

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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

i got 7, but it is a wrong answer
could you share your ways of solving such problems?
[Reveal] Spoiler: OA

Last edited by Bunuel on 05 Sep 2013, 01:59, edited 1 time in total.
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Re: How many positive integers less than 30 have no common prime [#permalink]

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Galiya 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

i got 7, but it is a wrong answer
could you share your ways of solving such problems?


Factorization of 30 = 2*3*5. Thus, we have to find the no of positive integers, which don't have any one of these as their factors. Thus, it is nothing but all the primes below 30 AND also the integer 1.

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

D.
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Re: How many positive integers less than 30 have no common prime [#permalink]

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New post 05 Sep 2013, 02:03
Galiya 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

i got 7, but it is a wrong answer
could you share your ways of solving such problems?


30=2*3*5. So, the number must be less than 30 and not have primes 2, 3, or 5.

This means that the number could be: 1, 7, 11, 13, 17, 19, 23, or 29. Total of 8 numbers.

Answer: D.
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Re: How many positive integers less than 30 have no common prime [#permalink]

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New post 05 Sep 2013, 02:57
but isnt 1 a universal factor for all integers?

i threw it away because thought so

and moreover we are asked "no common prime factor" 1 is not a prime factor, isnt it?
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Re: How many positive integers less than 30 have no common prime [#permalink]

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Galiya wrote:
but isnt 1 a universal factor for all integers?

i threw it away because thought so


We are looking for positive integers less than 30 which have no common prime factor with 30.

1 does not have any common prime factor with 30 (in fact 1 does not have any prime factor).

Does this make sense?
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Re: How many positive integers less than 30 have no common prime [#permalink]

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New post 17 Nov 2013, 06:13
Very conceptual question. Thanks for posting.
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How many positive integers less than 30 have no common prime [#permalink]

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This question is easily solved by using Euler’s totient function. It defines the number of integers co-prime to a given number.
Ɵ(n)=n(1-1/p)(1-1/q)(1-1/r) … etc.
Where p, q, r … are prime factors of a given number.
In our case 30=2*3*5
So Ɵ(30)=30*1/2*2/3*4/5=8
Answer D
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Re: How many positive integers less than 30 have no common prime [#permalink]

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New post 01 Mar 2017, 02:29
Prime number less than 30 which are not the prime factor of 30 : 1, 7,11,13,17,19,23,29. Total 8 As we can see that no two numbers can yield the number less than 30, therefore only these will be the numbers.Hence Option D
Re: How many positive integers less than 30 have no common prime   [#permalink] 01 Mar 2017, 02:29
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