The easiest way to keep them straight is to remember that a multiple is something that was mutliplied.
So if we were to compare 5 to 50, we would say that 50 is a multiple of 5 (because 5 times another number produces 50) and that 5 is a factor of 50 (because 50 broken down by another number produces 5)
So there you have it.
A factor is basically the result of one number being divided by a second number, and this second number is a factor of the first number.
As I discussed in my post about prime and composite numbers , 1 is the lowest factor of every positive number. The highest factor is the number itself.
To find factors, use some of the divisibility tests I mentioned before in my post about prime and composite numbers or take the easy way and just look below:
- Even Numbers: Any even number (other than 2) is divisible by 2. If it is odd, then it is not divisible by 2.
- Digital Roots: When you add up all the numbers of a digit, you get its digital root. If you get a multiple digit answer, then keep adding digits until you get a single digit. When you have a number with a digital root of 3, 6, or 9, it is divisible by 3. It only works for 3.
455 = 4+5+5 = 14 = 1+4 = 5 ---> not divisible by 3
- Starting with a plus sign, alternate plus and minus signs in front of all digits and then solve them. If the answer is 0 or any number that is divisible by 11, then the number you solved for is divisible by 11.
616 = +6-1+6 = 5 + 6 = 11 ---> divisible by 11
820 = +8-2+0 = 6 ---> not divisible by 11
- If a number ends in 5 or 0, it is divisible by 5.
Start with 1:
100*1 = 100
So far, we know that 1 is a factor of 100.
Next, move on to 2:
2*50 = 100
So now we add 2 to the list of factors of 100.
Try 4:
4*25 = 100
Try 6:
We know there is nothing that we can multiply by 6 that will give us a whole number.
6 goes into 100 16 times with a remainder of 4, so not a whole number.
6 is not a factor of 100.
Try 8:
Again, nothing that we can multiply by 8 will give us a whole number.
8 goes into 100 12 times with a remainder of 4, so not a whole number.
8 is not a factor of 100.
Try 10:
10*10 = 100
Try 20:
20*5 = 100
Try 50:
50*2 = 100
Try 100:
100*1 = 100
So here is our list of the factors of 100:
1, 2, 4, 10, 20, 50, 100
Prime Factorization
You can also make a factoring "tree" by breaking down a number into its prime factors, which is a group of prime numbers that, when multiplied together, result in that number. It is called a "tree" because the lines that you point outward for each number on top look like tree branches.
Factor trees are easy to do, all you do is start with two numbers that you know equal your original number and then break the resulting numbers down until they are prime numbers and can be broken down no further.
Let's use a factoring tree to break down 1000 into its prime factors:
1000
/ \
500 2 ---> prime factor
/ \
250 2 ---> prime factor
/ \
125 2 ---> prime factor
/ \
25 5 ---> prime factor
/ \
prime factor ---> 5 5 ---> prime factor
Taking all the prime factors that we ended up with, we can say that 2*2*2*5*5*5 = 1000. This is called decomposing a number down to its prime factors.
Think of prime factorization as a molecule that is made up of atoms. We are essentially breaking a molecule down to its atoms.
Greatest Common Factor
This method follows factoring into prime numbers in that finding the greatest common factor (GCF) involves finding the highest number that is a factor of every number in a set of numbers.
It is helpful to use a factoring tree to find the GCF.
Essentially, you will want to find the GCF for larger numbers with more factors, or when you are comparing more than two multiples.
Let's follow with our previous example, where we found the factors of 1000 (I added in 10, 20, 50, and 100 because they are factors; 10 * 100 = 1000 and 20 * 50 = 1000. We just didn't include them in our tree):
1, 2, 5, 10, 20, 25, 50, 100, 125, 250, 500, 1000
Now, let's say we are trying to find the GCF of 1000 and 1234
Let's break 1234 down using some factoring trees:
First of all, we know that 1234 is an even number, so it is divisible by 2.
1234
/ \
2 617
617 is an odd number, so let's try finding its digital root: +6-1+7 = 12...not divisible by 11
6+1+7 = 14 = 1+4 = 5...not divisible by 3
Looks like we've found a prime number!
Now we compare both factors:
Factors of 1000: 1, 2, 5, 10, 20, 25, 50, 100, 125, 250, 500, 1000
Factors of 1234: 1, 2, 617, 1234
So the largest number that both 1000 and 1234 share is 2. Therefore, 2 is the greatest common factor of 1000 and 1234!
Let's try finding the GCF of 1000 and 525.
We have already found the factors of 1000, so let's find the factors of 525:
525 525 525 525
/ \ / \ / \ / \
25 21 105 5 175 3 75 7
/ \ / \ / \ / \
5 5 3 7 7 15 35 5
Note how I broke 525 down into multiple factoring trees. I used my first tree, on the far left, as a template for the next few trees. For the first tree, I assumed that 525 was divisible by 25 (since 25 is a multiple of 5). When I got down to the prime numbers, I then knew that 525 was divisible by 3, 5, and 7, so I used those prime numbers to divide 525 by in the next three trees to the right. I did not continue to break these trees all the way down to their prime numbers because I already knew what those prime numbers were. I just needed to find the other factors of 525.
Again, we compare both factors:
Factors of 1000: 1, 2, 5, 10, 20, 25, 50, 100, 125, 250, 500, 1000
Factors of 525: 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525
The largest number that both 1000 and 525 share is 25. 25 is the greatest common factor.
You might be wondering, why not solve for the least common factor?
Well, the answer would always be 1, so there isn't much point.
Now try some problems to help cement your knowledge on GCFs.
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