Generating Factors Using Prime Numbers
& The Fundamental Theorem of Arithmetic
This method utilizes the unique prime factorization of a number to generate its factors. And the best part is that it doesn’t require any division, just basic math facts and multiplication. And given the opportunity, I always choose multiplication over division.
Recall in Lesson Twenty-five: Using Factor Trees to Find GCFs and LCMs we learned how to build factor trees and write out the prime factorization of a number. Please view that lesson first if you are unfamiliar with either of these concepts!
Example: Find All Factors of 72
Begin by drawing the factor tree for 72, circling primes as you go.
From there write out the prime factorization. Since we have three 2’s and two 3’s multiplied together, we can use exponents to group them.
Or we can keep them separated:
The Fundamental Theorem of Arithmetic
I should note that this idea that every number greater than 1 is either prime or capable of being rewritten as a product of prime numbers is a very important concept known as the Fundamental Theorem of Arithmetic.
Generating Factors of 72
By finding all combinations of these numbers we’ll generate all of the factors. I can begin by grouping one of the 2’s by itself and the rest of the numbers together to get the first pair of factors: 2, 36.
I could then group the first two 2’s together to obtain the factors 4 and 18.
Group the 2’s and 3’s respectively to obtain 8 and 9.
Group all the numbers together except for one 3 to obtain 24 times 3.
Lastly group one 2 and one 3 together to yield 6 times 12.
Once you have found all of the combinations, list the factors along with 1 and the number itself to have the complete list.
I find using the prime factorization is more efficient than dividing by trial because every combination is guaranteed to produce a factor pair. You simply have to make sure you’ve found them all.
I hope through the tutorials so far you are gaining a good sense of how numbers can be broken apart and put back together again. Once you have a solid understanding of numbers as components of a system adhering to a set of rules, you’ll begin to recognize shortcuts and patterns everywhere.
Next Lesson: The 1 and four 7’s Number Puzzle
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