What Are Prime numbers, GCF and LCM?
At this time, I also want to give you a couple more concepts that you’ll need throughout your math career.
A prime number is a number that is only divisible by 1 and itself: 2, 3, 5, 7, 11, 13, 17, and so on. Notice that no even numbers can be prime, except for 2. Every “regular” number (like 1, 2, 3, 4, and so on) can be factored into prime numbers. Prime numbers have exactly two factors.
A composite number is a number that is not a prime number. Composite numbers are 4, 6, 8, 9, 10, 12, and so on.
Note that the number “1” fits into its own category, and is neither prime nor composite. It is not prime, since it technically doesn’t have two factors.
Greatest Common Factor (GCF)
The Greatest Common Factor (or GCF) of numbers is just what it says: it’s the greatest number (other than 1) that goes into the numbers without any remainders. It can be one of the numbers, for example; if we have 3 and 6, the greatest common factor is 3, since 3 is the largest number that goes into both 3 and 6 perfectly. The Greatest Common Factor is sometimes called the Greatest Common Divisor.
To get the GCF of numbers, you can list all the prime factors of the numbers, match up factors on both sides, and multiply these to get the greatest factor.
Let’s try this for 12 and 18; see how we can “drill down” by starting with any two factors (you can always try to start with 2!). The order doesn’t make a difference; we will always get down to the prime factors! We divide each number down until we get prime numbers.
We’ll create what we call prime factor trees, since we’ll “dissect” the numbers and “branch out” and get down to the prime factors. The lowest factors (the ones that can’t be divided by any number except for itself and 1) are the lowest “leaves” on the tree and are the prime factors.
To get the GCF, we see what factors are on all sides, keep track of them by circling them, and then multiply them together (just on one side). Since we have a “2 x 3” circled on both sides, the GCF is 6 (2 x 3). Note that we don’t circle the other 2 under the 12, and other 3 under the 18, since we don’t have matches for them on the other side. But we can circle two of the same numbers if they match up on both sides.
So 6 is the factor (or number that goes into) 12 and 18 that is the largest. We’ll use the GCF later when we want to reduce a fraction to its “simplest” form.
Here is another example:
Find the GCF of 20, 28 and 56:
Create a factor tree and circle any matches of prime factors. Then we’ll multiply these together to get the GCF.
To get the GCF, we see what factors are under all numbers (factorizations), keep track of them by circling them, and then multiply them together (just on one side). Since we have a “2 x 2” circled under all numbers, the GCF is 4 (2 x 2). Note that we could circle two of the same numbers (the 2’s) since there were part of the factorization under all the numbers.
Least Common Multiple (LCM)
The Least Common Multiple (or LCM) is used more often, since we’ll see later that we’ll use it to add and subtract fractions. The Least Common Multiple is also called the Least Common Denominator (LCD), when used with fractions.
Let’s find the LCM of 4 and 6. To do this, we find the smallest number that they both go into (or that are multiples of the numbers). We can begin by writing down all the numbers they go in to, starting with the actual number:
MULTIPLES of 4: 4, 8, 12, 16, 20, 24, 28, 32 …
MULTIPLES of 6: 6, 12, 18, 24, 30 …
Then we find the lowest number that is in both lists. Note that the smallest multiple of both the numbers is 12. Another common multiple is 24, but this not the smallest. You can always get a common multiple of numbers (you can have more than two!) by multiplying them together, but this is not always the smallest one.
We can also use a factor tree to find the LCM. For each prime factor, we find where it occurs the most often under a number (even if it’s only once) and circle these factors. Note that if we have circled one or more prime factors, we don’t circle it again under any other number.
Then we multiply them all across to get the LCM.
Then we multiply all the factors that we’ve circled, and we get 2 x 2 x 3 = 12. So the LCM of 4 and 6 is 12.
Now let’s find the LCM of 14, 18 and 168 by using factor trees.
Again, we list the prime factors across all numbers, and for each prime factor, we circle them if and only if they occur the most often under that number, and don’t repeat circling factors across numbers.
Make sure all the prime factors are covered (7, 2, and 3); it looks like they are. Also note that we only circled the 7 under one of the numbers (14).
Then we multiply everything that’s circled, and we get 7 x 3 x 3 x 2 x 2 x 2 = 504. So the LCM of 14, 18, and 168 is 504.
Learn these rules and practice, practice, practice!