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greatest common factor

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Manager
Joined: 30 Dec 2008
Posts: 121

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16 Jan 2009, 00:29
The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
a. 3
b. 14
c. 30
d. 42
e. 70

thorough explanation would be appreciated..

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Intern
Joined: 05 Jan 2009
Posts: 9

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16 Jan 2009, 01:00
d.42

16= 2^4. Since the GCF between 16 and n is 4 we know that n has two '2' as factors
45= 3^2x5. Since the GCF between 45 and n is 3 we know that n has one '3' and do not have any '5' as factors.

210= 2x3x5x7
Let's check these 4 factors:
'2': the GCF between n and 210 will have only one (n has two but 210 has one).
'3': the GCF between n and 210 will have only one (both have one)
'5': the GCF between n and 210 will NOT have 5. (n does not have)
'7': We do not know if this factor will be in the GCF because it has not apeear so far. But certainly could be a factor.
So one of the possible solutions is: 2x3x7=42
Manager
Joined: 30 Dec 2008
Posts: 121

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16 Jan 2009, 02:43
Thank you! that's the right answer.
what is the approach here?
are we trying to find the smallest prime factor and then just multiply them out?
is this the rule? Is there an easy logic to deal with this kind of problem?
I think I'm kinda weak in factors, prime, GCF, remainder/divisibility properties.

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Re: greatest common factor   [#permalink] 16 Jan 2009, 02:43
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greatest common factor

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