Last Updated on April 26, 2023
“Factor” is a term you may have come across while prepping for your GRE. For example, perhaps you were asked to break a number down into its factors or prime factors, maybe even to determine the total number of factors of the number. We’ll get some practice with these kinds of questions in this article, but let’s first discuss what exactly a factor is.
What is a Factor?
A factor is a number that divides evenly into another number, leaving no remainder. More formally, we can say that if x and y are integers and x/y = integer, then y is a factor of x. To practice this idea, let’s look at the factors of a couple of numbers. For instance, suppose we were asked the following:
What are the factors of 12?
We can see that 1, 2, 3, 4, 6, and 12 are all factors of 12 because they all divide evenly into 12 with no remainder.
What are the factors of 18?
We can see that 1, 2, 3, 6, 9, and 18 are all factors of 18 because they all divide evenly into 18 with no remainder.
If you are struggling to quickly determine all the factors of a particular number, you can perform the following steps.
Steps to determine all the factors of a particular number
As an example, let’s determine the factors of 20.
Step 1: Begin with 1 and the number itself. These are factors of every integer. Thus, the first two factors of 20 are 1 and 20. Note that 1 and 20 both evenly divide into 20 because 1 times 20 is 20.
Step 2: Count up from 1, checking whether each integer is a factor of the given number. Stop as soon as you reach a repeated factor.
The next integer after 1 is 2. Since 20/2 = 10, we can say that 2 divides evenly into 20 and that 2 and 10 are factors of 20. Note that 10 also divides evenly into 20 because 20/10 = 2.
Next, we check 3. Since 3 does not divide evenly into 20, we can say that 3 is not a factor of 20.
Moving on to 4, since 20/4 = 5, we can say that 4 and 5 are factors of 20.
Since 20/5 = 4, we can say that 5 and 4 are factors of 20. However, since we have now found repeated factors, we can stop.
The factors of 20 are 1, 2, 4, 5, 10, and 20.
KEY FACT:
You can stop looking for the factors of a given number as soon as you reach a repeated factor.
Now, let’s discuss prime factorization.
Prime Factorization
We now understand how to find all of the factors of a particular number. However, we also need to know how to break a number down into its prime factors. This process is called “prime factorization.” As we may recall, a prime number is an integer greater than 1 that has no factors other than 1 and itself. For example, 2, 3, 5, and 7 are all prime numbers because the only numbers that will divide evenly into them are 1 and themselves. A simple way to find the prime factors of a number is with a factor tree. Let’s use the number 56 as an example.
KEY FACT:
A prime number is an integer greater than 1 that has no factors other than 1 and itself.
To start the factor tree, we need any factor of 56. If we have a handle on our multiplication tables, we may recall that 56/8 = 7. Thus, we will split 56 into 8 times 7. Since 7 is a prime number, it cannot be factored any further, and so we leave it alone. Next, we break 8 into its factors of 4 and 2. Since 2 is a prime number, we leave it alone. To finish the tree, we factor 4 into 2 times 2. The factor tree is shown below:
After seeking out the prime factors present in the factor tree, we can thus say that the prime factorization of 56 is 2 ✕ 2 ✕ 2 ✕ 7. Note that we can check our prime factorization by multiplying this string of numbers to ensure we get back the number we started with: 2 ✕ 2 ✕ 2 ✕ 7 = 4 ✕ 2 ✕ 7 = 8 ✕ 7 = 56.
TTP PRO TIP:
A factor tree is a useful way of determining the prime factorization of an integer.
Let’s practice with one more prime factorization. Let’s break 96 into its prime factors. Since 96 is even, we know that it’s divisible by 2. Thus, we can break 96 into 2 times 48. Since 2 is a prime number, it cannot be factored any further, and so we leave it alone. We can next break 48 into 2 times 24. Next, we can break 24 into 8 times 3. Next, we can break 8 into 4 times 2, and finally, 4 is broken into 2 times 2. Thus, the factor tree of 96 looks like the following:
Seeking out the prime factors in the factor tree, we see that the prime factorization of 96 is 2 ✕ 2 ✕ 2 ✕ 2 ✕ 2 ✕ 3.
Let’s now practice with a couple of examples.
Prime Factorization Example 1
294 has how many distinct prime factors?
- 1
- 2
- 3
- 4
- 5
Solution:
To determine the number of distinct prime factors of 294, we can prime factorize 294. Since 294 is an even number, we know that it’s divisible by 2. Thus, we can factor 294 as 2 times 147.
Next, since the digits of 147 sum to a multiple of 3, we know that 147 is divisible by 3. Thus, we can factor 147 into 3 times 49.
Lastly, 49 can be factored as 7 times 7. We can express all of this in a factor tree:
In the factor tree above, we see that 294 has the prime factors 2, 3, and 7. Thus, 294 has 3 distinct prime factors.
Answer: C
Prime Factorization Example 2
What is the sum of all the prime factors of 120?
- 12
- 14
- 17
- 19
- 12
Solution:
Let’s prime factorize 120. Since 120 is an even number, we know that it’s divisible by 2. Thus, we can factor 120 into 2 times 60.
Next, we can factor 60 as 3 times 20.
We can factor 20 as 5 times 4.
Lastly, 4 can be factored as 2 times 2. We can express all of this in a factor tree:
We see that the prime factors of 120 are 2, 2, 2, 3, and 5. Thus, the sum of those prime numbers is 2 + 2 + 2 + 3 + 5 = 14.
Answer: B
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