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How many factors greater than 1 do 120, 210, and 270 have in

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Retired Moderator
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GRE 1: Q167 V156
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How many factors greater than 1 do 120, 210, and 270 have in [#permalink] New post 12 Aug 2018, 15:31
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Question Stats:

36% (02:35) correct 64% (02:13) wrong based on 25 sessions
How many factors greater than 1 do 120, 210, and 270 have in common?

(A) One
(B) Three
(C) Six
(D) Seven
(E) Thirty
[Reveal] Spoiler: OA

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Retired Moderator
User avatar
Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 173

Kudos [?]: 2977 [0], given: 394

Re: How many factors greater than 1 do 120, 210, and 270 have in [#permalink] New post 15 Aug 2018, 05:14
Expert's post
Explanation

Pick one of the numbers and list all of its factors. The factors of 120 are: 1 & 120, 2 & 60, 3 & 40, 4 & 30, 5 & 24, 6 & 20, 8 & 15, 10 & 12. Since the problem specifically asks for factors “greater than 1,” eliminate 1 now. Now cross off any factors that do not go into 210:

Attachment:
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Now cross off any factors remaining that do not go into 270. Interestingly, all of the remaining factors (2, 3, 5, 6, 10, 15, 30) do go into 270. There are 7 shared factors.
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Re: How many factors greater than 1 do 120, 210, and 270 have in [#permalink] New post 04 Apr 2020, 22:31
This is the other way to do it
120 = 10 * 12 = 2*5*3*2
210 = 3*7*2*5
270 = 3*3*3*2*5

Find common prime factors in all three numbers above: 2^1 * 3^1 * 5^1. So to find all factors add 1 onto each exponent and multiply altogether (1+1)*(1+1)*(1+1) = 8. But we don't want 1. So 8 - 1 = 7
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Re: How many factors greater than 1 do 120, 210, and 270 have in [#permalink] New post 05 Apr 2020, 04:20
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factors of 120 = 2*2*5*3
factors of 210 = 2*5*3*7
factors of 270 = 3*3*3*2*5
the factors common to all is 2,3,5
no of ways of selecting one out of the 3 common factors is 3C1 ( in other words, we can select 2 or 3 or 5)
no of ways of selecting two common factors out of 3 is 3C2 ( in other words we can select 2*3 or 2*5 or 3*5)
no of ways of selecting all three common factors is 3C3 ( in other words we can select 2*3*5)
all common factors greater than 1 is 3C1 or 3C2 or 3C3 = 3C1 + 3C2 + 3C3 =7
hence ans is D
Re: How many factors greater than 1 do 120, 210, and 270 have in   [#permalink] 05 Apr 2020, 04:20
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