Presenting the detailed explanation of the solution.
GivenWe are given two numbers 180 & 96 and are asked to find their number of common positive factors.
ApproachThe 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
Working OutPrime 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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