Understanding prime number rules and properties is important for solving various GMAT quantitative problems involving factors, multiples, divisibility, and number properties. Not only are prime numbers inherently fascinating from a mathematical perspective, but they also appear frequently in GMAT problems, even when the term “prime” isn’t explicitly mentioned.

Familiarizing yourself with these concepts will help you tackle such questions more efficiently and accurately.

Let’s dive in and explore the key prime number rules you need to know for GMAT prime numbers questions.

## Rule 1: Definition of a Prime Number

A prime number is a positive integer with exactly two factors: 1 and itself. It’s important to note that 1 is not considered a prime number because it only has one factor (itself). The smallest prime number is 2, which is also the only even prime number. All other prime numbers are odd.

## Rule 2: Identifying Prime Numbers

To determine whether a number is prime, check if it has two positive divisors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7. However, 12 is not prime because it has factors like 2, 3, 4, and 6 in addition to 1 and 12.

This is how one of our instructors explains it:

“*If you’ve gone through school, you’ve likely heard the definition of a prime as “any number that can be divided only by 1 and itself.” Or, put differently, “any number that has only 1 and itself as factors.” For example, 3 is a prime number because 1 and 3 are the only numbers that are factors of 3.*

*However, there is something slightly problematic here. I always ask my students, “Okay, well then, is 1 prime? 1 is divisible by only 1 and itself.” Many people are under the misconception that 1 is a prime number, but in truth, 1 is not prime. *

*There is a better way to think about prime numbers:*

*A prime number is any number that has EXACTLY TWO FACTORS.*

*By that definition, 1 is not prime, as it has only a single factor.* “

A *composite number*, on the other hand, has more than two factors. For example, 6 is a composite number because it can be divided by 1, 2, 3, and 6. It is composed of at least two prime number (2*3). In contrast, the number 5, which can only be factored into 1 and 5, is a prime number.

## Rule 3: The Uniqueness of 2

The number 2 is the only even prime number. This is because any other even number will have at least three factors: 1, 2, and itself. Since prime numbers must have exactly two factors, no other even number can be prime. In other words, even really just means “divisible by 2”, and so anything besides 2 that is divisible by 2 won’t be prime.

This is how one of our instructors explains it:

*“Any other even number must have more than two factors because apart from 1 and the number itself, 2 must also be a factor. For example, the number 4 will have 1 and 4 as factors, of course, but it will also have 2 since it is even. No even number besides 2, therefore, will have exactly two factors. *

*Another way to read this, then, is that every prime number other than 2 is odd.”*

## Rule 4: Memorizing Prime Numbers

While it’s not necessary to memorize a large list of prime numbers, knowing the prime numbers up to a certain value is helpful.

For the GMAT, you should be familiar with the following primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37. You can also remember 41 and 43.

As already mentioned, it’s good to know up to a certain value, but it’s unnecessary to go beyond that into conspicuously larger numbers because the GMAT, as a test, is less interested in your ability to memorize large primes and more interested in your reasoning skills and your ability to draw conclusions about novel problems on the fly. If you know the following, you should be set.

(It’s interesting that many people forget that 27 is actually not prime. But don’t beat yourself up if this happens to you: Terence Tao, one of the world’s leading mathematicians and an expert on prime numbers, actually slipped briefly on national television once and said 27 was prime before catching himself. And he’s one of the best in the world. So even the best of the best make these mistakes.)

## Rule 5: Prime Factorization

Every positive integer can be uniquely expressed as a product of prime numbers, known as its prime factorization. To find a number’s prime factorization, **divide it by the smallest prime factor **possible and repeat the process until you’re left with only prime factors.

For example, let’s find the prime factorization of 180:

180 ÷ 2 = 90 (2 is the smallest prime factor of 180)

90 ÷ 2 = 45

45 ÷ 3 = 15 (3 is the smallest prime factor of 45)

15 ÷ 3 = 5

5 is prime, so we stop here.

Therefore, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5 = 22 × 32 × 5.

Understanding prime factorization is crucial for solving various GMAT problems, such as finding the number of factors, determining the Greatest Common Factor (GCF) or Least Common Multiple (LCM) of two or more numbers, and simplifying fractions.

## Rule 6: The Sieve of Eratosthenes

The Sieve of Eratosthenes is an efficient method for finding all prime numbers up to a given limit.

It works by iteratively marking the multiples of each prime number, starting with 2 and then moving on to the next unmarked number. The unmarked numbers that remain at the end of the process are prime.

Here’s how the algorithm works:

- Create a list of all integers from 2 up to the desired limit.
- Start with the first prime number, 2, and mark all its multiples (excluding itself) as composite.
- Move to the next unmarked number, which will be the next prime number, and repeat step 2.
- Continue this process until you’ve marked all multiples of primes up to the square root of the limit.
- The remaining unmarked numbers are prime.

While you won’t be asked to perform the Sieve of Eratosthenes on the GMAT, understanding the concept can help you grasp the distribution and properties of prime numbers. Watch this video for more tips on prime numbers.

## Rule 7: The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic, also known as the Unique Factorization Theorem, states that *every positive integer greater than 1 can be **uniquely** represented as a product of prime numbers up to the order of the factors*.

This means that no matter how you factor a number into primes, you’ll always get the same set of prime factors (although the order may differ).

For instance, consider the number 1,260. We can factor it as:

1,260 = 2² × 3² × 5 × 7

No matter how we factor 1,260, we’ll always get the same set of prime factors: two 2s, two 3s, one 5, and one 7. This unique representation is crucial for solving problems involving factors, divisibility, and number theory on the GMAT.

This theorem emphasizes the importance of prime numbers as the building blocks of all integers.

## Rule 8: Applying Prime Number Rules on the GMAT

When tackling GMAT problems involving prime numbers, remember these rules. Look for opportunities to break down numbers into their prime factors, determine the number of factors a given integer has, or identify patterns related to prime numbers.

Practice applying these rules to various problem types to strengthen your understanding and problem-solving skills.

By mastering these prime number rules and understanding their implications, you’ll be well-equipped to handle any GMAT question that involves primes. Remember, the GMAT is more concerned with your ability to reason and draw conclusions than your memorization skills.

Focus on internalizing these rules and applying them flexibly to novel problems, and you’ll be primed for success on the GMAT!

## Frequently Asked Questions on Prime Numbers

### 1. How to find prime numbers?

One way to find prime numbers is to use factorization. Prime factorization involves breaking down a number into its prime factors, which are the smallest divisors of the number that are also prime.

To find prime numbers using factorization, start with the smallest prime number, 2, and divide the given number by 2 as many times as possible. Then, move on to the next prime number, 3, and repeat the process.

Continue this process with subsequent prime numbers until the quotient itself becomes a prime number. The prime factors found during this process, along with the final prime quotient, constitute the prime factorization of the original number. Numbers that cannot be factored further (other than 1 and themselves) are prime numbers.

The prime numbers from 1 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### 2. How many prime numbers are there?

There are infinitely many prime numbers, as Euclid proved around 300 BCE. His proof demonstrates that if there were a largest prime number, multiplying all the primes up to that number and adding 1 would result in a new prime or a number divisible by a larger prime, contradicting the assumption that the largest prime had been found. Ponder this to see if you can make it intuitive for yourself.

### 3. What is the difference between prime and composite numbers?

Prime numbers have exactly two factors: 1 and themselves. Composite numbers, on the other hand, have more than two factors. In other words, composite numbers can be divided evenly by at least two other prime numbers.

### 4. How many prime numbers are between 1 and 100?

There are 25 prime numbers between 1 and 100. These primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

### 5. Why are prime numbers important?

Prime numbers are the building blocks of all positive integers, as every number can be uniquely represented as a product of primes (Fundamental Theorem of Arithmetic). They play a crucial role in number theory, cryptography, and computer science. In cryptography, large prime numbers are used to create secure encryption keys, ensuring the safety of sensitive data and communications.

## Prime Numbers Exercise:

Here are two Official Guide problems that take the basics of Prime Numbers and force you to do a little reasoning. *Give them a shot:*

- What is the smallest integer n for which 25^n > 5^12?

(A) 6

(B) 7

(C) 8

(D) 9

(E) 10

- How many prime numbers between 1 and 100 are prime factors of 7,150?

(A) One

(B) Two

(C) Three

(D) Four

(E) Five

Do you find prime number rules overwhelming? Working with a private GMAT tutor can help you understand them and ace the GMAT!