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How many positive factors do 180 and 96 have in common?

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How many positive factors do 180 and 96 have in common?  [#permalink]

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New post Updated on: 06 May 2015, 10:52
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How many positive factors do 180 and 96 have in common?

A. 6
B. 12
C. 16
D. 18
E. 24

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Originally posted by Baten80 on 06 May 2015, 10:50.
Last edited by Bunuel on 06 May 2015, 10:52, edited 1 time in total.
RENAMED THE TOPIC.
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How many positive factors do 180 and 96 have in common?  [#permalink]

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New post Updated on: 12 Jun 2015, 02:42
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Presenting the detailed explanation of the solution.

Given
We are given two numbers 180 & 96 and are asked to find their number of common positive factors.

Approach
The question talks about finding the common positive factors of two numbers. The common factors should have its prime numbers in the power which is present in both the original numbers. Hence our first step would be to prime factorize the original numbers.

Once we prime factorize the number we can look at the common prime factors of the number along with their powers to find out the common factors.

For example, take two numbers 6 and 12. \(6 = 2*3\) and \(12= 2^2 * 3\). Both the numbers have the common prime factors of 2 & 3 but their power differs.Since 12 has \(2^2\) in it and 6 has \(2^1\) in it, the common factors of 6 & 12 can't have more than \(2^1\) in it. Similarly, \(3^1\) would be the maximum power of 3 in the common factors of 6 & 12.

So, the common factors of 6 & 12 will be by the combination of \(2^1\) and \(3^1\). There can be 2 * 2 = 4 possible combinations which are ( \(2^0 * 3^0, 2^1*3^0, 2^0 * 3^1, 2^1* 3^1\)) i.e. 1, 2,3 and 6.


You can also observe here that 6 is the HCF(6, 12) and the common factors of 12 & 6 are the factors of the HCF(6, 12) = 6. This is possible because we chose the lowest powers of the common prime numbers, the same way we do in finding the HCF of a set of numbers :-D

Working Out
Prime factorizing 180 & 96 would give us \(180 = 2^2 * 3^2 * 5\) and \(96 = 2^5 * 3\).

From the above prime factorization of 180 & 96, we can analyze that the common factors of 180 & 96 would come from a combination of ( \(2^2\) and \(3^1\)). Thus there are 3 * 2 = 6 possible combinations i.e. 6 common factors of 180 & 96.

The common factors would be (\(2^0*3^0\), \(2^1 * 3^0\), \(2^0*3^1\), \(2^2*3^0\), \(2^1*3^1\), \(2^2 * 3^1\)) i.e. 1,2,3,4,6 & 12 which are the factors of HCF(180,96) = 12


One of the common mistakes people make in a LCM-GCD question is not having Prime Factorization as their default approach. Go through the 2nd pitfall in our article 3 Deadly Mistakes you must avoid in LCM-GCD questions to know more about the power of prime factorization in solving LCM-GCD questions.


Hope its clear!

Regards
Harsh
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Originally posted by EgmatQuantExpert on 07 May 2015, 23:28.
Last edited by EgmatQuantExpert on 12 Jun 2015, 02:42, edited 1 time in total.
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Re: How many positive factors do 180 and 96 have in common?  [#permalink]

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New post 07 May 2015, 20:28
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The number of common factors will be same as number of factors of the Highest Common Factor(HCF)
HCF of 180 and 96 is 12
Number of factors of 12 = 6

Answer : A

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Re: How many positive factors do 180 and 96 have in common?  [#permalink]

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New post 08 May 2015, 02:40
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Re: How many positive factors do 180 and 96 have in common?  [#permalink]

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New post 30 Jun 2019, 07:02
Bunuel wrote:
Baten80 wrote:
How many positive factors do 180 and 96 have in common?

A. 6
B. 12
C. 16
D. 18
E. 24


To find the number of common factors of two integers:
1. Find the greatest common divisor (GCD) of the two integers;
2. Find the number of factors of that GCD.

GCD of 180 and 96 is 12 = 2^2*3. The number of factors of 12 is (2 + 1)(1 + 1) = 6.

Answer: A.


I did in the exact same way. Finding out the GCD and finding the number of factors for GCD. hence A
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Re: How many positive factors do 180 and 96 have in common?   [#permalink] 30 Jun 2019, 07:02
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