The first post in this mini-series addressed factoring using multiplication facts and the rules of divisibility. Today, we’re going to factor a number into its primes.
A prime number is any number that can be divided only by itself and 1. Some people find this a more useful way to find common denominators,and it can actually make simplifying radical expressions much simpler.To start factoring a number to its primes, we need to either apply the multiplication facts or the rules of divisibility to it.
Let’s use 72 for this example. 72 is 8 * 9.
72
/\
8 9
Neither 8 nor 9 is prime, so we’re going to factor both of them. We know that 8 is 4 * 2; and 9 is 3 * 3, so let’s add those to our factor tree.
72
/ \
8 9
/\ /\
2 4 3 3
Now we’re getting somewhere! Both 2 and 3 are prime, leaving us only the 4 to factor. The factor tree looks like this now.
72
/ \
8 9
/ \ /\
2 4 3 3
/\
2 2
I’ve bolded the prime numbers at the end of each branch so we can see them clearly. The prime factorization of 72 is 2 * 2 * 2 * 3 * 3.
If you use this method to find GCFs, then you’ll need to find all the primes both numbers have in common and multiply them back together. For example, if you were comparing 72 to 12, you’d find that both numbers have two 2s and a 3 in their list of prime factors. 2 * 2 * 3 equals 12, so 12 would be the GCF for 12 and 72.
[...] Prime factorization is probably the quickest, most foolproof way to determine the GCF or LCM for two numbers. In fact, prime factorization gives you everything you need to find both the GCF and the LCM in a single bit of work. [...]
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