Why Do All Factors of a Number Arise from Combinations of Its Prime Factors?

[jman115]

When I teach GCF to students, I show them how to find via the prime factorization and explain to them how the PF can get you all the factors of a number by multiplying different combinations of the Prime Factors and then proceed to explain why they are supposed to multiply the common Prime factors for the gcf.
My question is, why does multiplying different combinations of the prime factors get you ALL of the number's factors?

 

[masters1]

When I teach GCF to students, I show them how to find via the prime factorization and explain to them how the PF can get you all the factors of a number by multiplying different combinations of the Prime Factors and then proceed to explain why they are supposed to multiply the common Prime factors for the gcf.
My question is, why does multiplying different combinations of the prime factors get you ALL of the number's factors?

Hi jman115,

I know you know this already, but every composite number can be factored into the products of only prime numbers. Any combination of products with these prime factors will yield a composite factor of the original number.

Don't know if that's answers your question. Hope so.

 

[jman115]

"Any combination of products with these prime factors will yield a composite factor of the original number." I stated this fact in my opening thread.

I am asking why this works. When you multiply all combinations of the prime factors you get all the composite factors of that number. I want to know why this works.

 

[Jameson]

This is a nice visual demonstration from Wikipedia of the prime factorization process. Any composite factor of the original number will be broken down into its own product prime factors, which are part of the original number's prime factor list.

View attachment 31

Take a number like 64. This could be broken down into 32*2 or 16*4, then repeated until you have only the prime factors. No matter which way you break down a number into composite factors then into prime factors, the end result will be the same list of prime factors. Because the list of prime factors is the same no matter which composite factors you start with, some combination of prime factors multiplied together will also produce any given composite factor.

 

[CellCoree]

This is a great question! The reason why multiplying different combinations of the prime factors gets you all of the factors of a number is because of the fundamental theorem of arithmetic. This theorem states that every positive integer can be expressed as a unique product of prime numbers. So, by finding the prime factors of a number and multiplying them in different combinations, we are essentially breaking down the number into its prime factors and then recombining them in different ways. This ensures that we are accounting for all possible factors of the number. Furthermore, by multiplying the common prime factors for the GCF, we are essentially finding the highest common factor that all the different combinations of prime factors have in common. This is why it is a useful method for finding the GCF of a number. I hope this explanation helps to clarify the concept for you and your students!