Is 1 a Prime Number?

The short answer to the burning question: “Is 1 a prime number?” is no. The number 1 is not a prime number. In this article, we’ll take a quick look at why this is so. But more importantly, we’ll look at several sample GMAT questions that you could get wrong if you mistakenly thought that 1 is 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.

Here are the topics we’ll cover:

• Prime and Composite Numbers
• Prime Factorization
  • Example 1: Sum of Prime Factors
  • Example 2: Number of Prime Factors
• Key Takeaways
• What’s Next?

Let’s first review some definitions.

 Prime and Composite Numbers

Consult any math textbook, and you will learn that a prime number is an integer greater than 1 with two positive factors: 1 and itself.

KEY FACT:

A prime number has only two positive factors: 1 and itself. Therefore, prime numbers are divisible by only two positive numbers.

You might argue that 1 x 1 = 1, and so it fits the definition of a prime number. But be careful! 1 has only ONE factor: 1. In order for a number to be prime, the two factors must be different from each other. Since 1 x 1 = 1 has only one factor (number 1), the definition of a prime number doesn’t fit this situation, and so 1 is not a prime number.

Now that we 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. We extend the list to the first 5 prime numbers: 2, 3, 5, 7, and 11.

KEY FACT:

The number 2 is the only even prime number and the smallest prime number.

Composite numbers are whole numbers that have more than 2 positive 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:

Composite numbers have more than 2 positive factors.

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

Any composite number can be expressed as a product of prime numbers. For example, 20 = 2 x 2 x 5 = 2^2 x 5. This prime factorization is easy to do, but finding the prime factorization of larger numbers can be challenging. Let’s look at a process that makes this task less arduous.

If we ever need to see which prime numbers comprise a particular number, we follow the process of prime factorization, also called prime decomposition. An easy way to perform 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 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: Sum of Prime Factors

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 (incorrect) sum would be 16 instead of 15.)

Answer: D

Example 2: Number of Prime Factors

285 is comprised of how many different prime factors?

A. 3
B. 4
C. 5
D. 6
E. 7

Solution:

As we can see, 285 = 3 x 5 x 19, so 285 is composed of 3 different prime numbers.

Answer: A

Key Takeaways

In this article, the key facts that we have covered are the following:

• A prime number has two positive factors: itself and 1.
• The number 1 is not a prime number
• The number 2 is the only even prime, and it is the smallest prime.
• A composite number has more than two positive factors.
• The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
• Prime factorization is a technique used to determine the prime factors of any composite number.

What’s Next?

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.