Last Updated on October 8, 2023
The number 1 is not a prime number. Because of the definition of a prime number and the function of the number 1 in number theory, which doesn’t allow 1 to be a prime number for mathematical reasons, 1 is not prime. The good news is that, on the GMAT, you do not need to know why one is not a prime number; you just need to memorize the fact that it is not.
Let’s review the definition of a prime number:
A prime number is an integer greater than 1 that has no factors other than 1 and itself.
KEY FACT:
A prime number has only two factors: 1 and itself. Therefore, prime numbers are divisible by only two numbers.
Since we already know that 1 is not a prime, we see that the first prime number (and the only even prime number) is 2, since the factors of 2 are 2 and 1. The first 5 prime numbers are 2, 3, 5, 7, and 11.
Composite numbers are whole numbers that have more than 2 factors. Some examples of composite numbers are 6 (because 6 has factors of 1, 2, 3, and 6) and 8 (because 8 has factors of 1, 2, 4, and 8). Note that no prime number is a composite number.
KEY FACT:
The number 2 is the only even prime number and the smallest prime number.
For the GMAT, you should memorize the first 25 prime numbers, so that if you are finding the prime factorization of a given number (more on this below), you will know when the factorization is complete.
Here is a list of prime numbers:
TTP PRO TIP:
Memorize the first 25 prime numbers.
Let’s now discuss one prominent way that prime numbers are used on the GMAT: prime factorization.
Prime Factorization
If we ever need to see which prime numbers make a particular number, we follow the process of prime factorization. An easy way to find the prime factorization of a number is to find a number that evenly divides the original number and keep factoring the quotients until we are left with just prime numbers. A useful way to stay organized and efficient when doing prime factorization is to use a factor tree. For example, let’s find the prime factorization of 160 by using the following factor tree:
Notice that we first split 160 into 16 and 10. Next, we split 16 into 8 and 2 and split 10 into 5 and 2. Then we split 8 into 4 and 2. Finally, we split 4 into 2 and 2. Once the factor tree is complete, we find all the prime numbers in the factor tree in order to rewrite 160 in prime factorization form:
160 = 2 x 2 x 2 x 2 x 2 x 5
160 = (2^5)(5^1)
Notice that 2 is raised to the fifth power because there are five twos, and five is raised to the first power because there is one five. Also, notice that 1 is not included in our prime factor tree because 1 is not a prime number!
TTP PRO TIP:
To find the prime factorization of a number, find a number that evenly divides the original number and keep factoring the quotients until you are left with just prime numbers.
Let’s now practice prime factorization with some examples.
Example 1:
What is the sum of all the prime factors of 180?
A. 10
B. 12
C. 13
D. 15
E. 16
Solution:
To determine the prime factors of 180, we can use a factor tree:
As we can see, 180 = 2 x 2 x 3 x 3 x 5. Here is the sum of those prime factors:
2 + 2 + 3 + 3 + 5 = 15
(Note that if you erroneously think that 1 is a prime number, you might include it as one of the prime factors of 180, and your sum would be 16 instead of 15.)
Answer: D
Example 2:
285 is made up of how many different prime factors?
A. 3
B. 4
C. 5
D. 6
E. 7
Solution:
As we can see, 285 = 5 x 57 = 5 x 19 x 3, so 285 is composed of 3 different prime numbers.
Answer: A
The topic of prime numbers is only one of many quant topics tested on the GMAT. For some general advice on how to improve your quant skills, check out our article about how to improve your GMAT quant score.
