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How many pairs have one or more prime factors common

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How many pairs have one or more prime factors common [#permalink]

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New post 12 May 2014, 14:02
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How many pairs (n, n+1) have one or more prime factors common, where n is an integer and 2 ≤ n ≤9 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Can someone have a go at this question and explain how to work this problem. The explanation simply states
[Reveal] Spoiler:
that this is a generalization of an elementary theorem in number theory.
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 May 2014, 01:04, edited 1 time in total.
Edited the question.

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Re: How many pairs have one or more prime factors common [#permalink]

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New post 12 May 2014, 14:15
Prime numbers between 2 and 9: 2, 3, 5, 7.
Each of this number isn't a factor of him+1, as simple as that.

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Re: How many pairs have one or more prime factors common [#permalink]

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New post 12 May 2014, 19:19
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Quickly look at the pairs
(2,3) - nothing common
(3,4) - nothing common
(4,5) - nothing common
(5,6) - nothing common
(6,7) - nothing common
(7,8) - nothing common
(8,9) - nothing common
(9,10) - nothing common

Hence A
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Re: How many pairs have one or more prime factors common [#permalink]

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New post 13 May 2014, 01:06
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Expert's post
dhirajx wrote:
How many pairs (n, n+1) have one or more prime factors common, where n is an integer and 2 ≤ n ≤9 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Can someone have a go at this question and explain how to work this problem. The explanation simply states
[Reveal] Spoiler:
that this is a generalization of an elementary theorem in number theory.


n and n+1 are consecutive integers.

Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.

Answer: A.

Hope it's clear.
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Re: How many pairs have one or more prime factors common [#permalink]

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New post 13 May 2014, 12:58
Solved this fairly easily by writing out the primes for 2 thru 9, and noticed no consecutive integers shared any prime factors...

2 - 2
3 - 3
4 - 2,2
5 - 5
6 - 2,3
7 - 7
8 - 2,2,2
9 - 3,3

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Re: How many pairs have one or more prime factors common [#permalink]

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Re: How many pairs have one or more prime factors common [#permalink]

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New post 11 Aug 2017, 10:03
consecutive numbers have no prime factors in common

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How many pairs have one or more prime factors common [#permalink]

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New post 12 Aug 2017, 00:05
Rule is that consecutive integers will not have common factors except 1 (and 1 is not a prime number).

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How many pairs have one or more prime factors common   [#permalink] 12 Aug 2017, 00:05
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