What Are Prime Numbers? Definition and Examples

What Are Prime Numbers?

A prime number is a fundamental building block in mathematics. Every whole number greater than $1$ is either prime or can be broken down completely into prime components. If a number has any divisors other than $1$ and itself, it is not prime.

For example, $13$ is a prime number because you can only divide it evenly by $1$ and $13$. The number $14$, however, is divisible by $1$, $2$, $7$, and $14$. Because it has four positive divisors, $14$ is not prime.

The list of prime numbers from $1$ to $20$ is: $2$, $3$, $5$, $7$, $11$, $13$, $17$, and $19$.

Key Ideas and Vocabulary

To master primality, you must confidently understand the difference between `factors vs multiples`.

• Factors: The whole numbers you multiply together to get a target number. For example, the `factors` of $10$ are $1$, $2$, $5$, and $10$.
• Multiples: The results of multiplying a target number by an integer. The `multiples` of $10$ are $10$, $20$, $30$, and so on.
• Composite numbers: Any whole number greater than $1$ that has more than two factors.

When you compare `prime vs composite numbers`, a prime number stands alone with exactly two factors. In contrast, `composite numbers` always contain three or more factors.

Visual Explanation

You can visualize prime numbers by arranging them into rectangular grids or arrays. A prime number can only form a single straight line. A composite number can be arranged into multiple different rectangles.

Worked Examples

You can identify whether a number is prime by methodically testing divisibility rules.

Example 1: Identifying a small prime number

Question: Is the number $27$ prime or composite?

Method:

1. Check for factors other than $1$ and $27$.
2. Test divisibility by $2$ and $5$. The number ends in $7$, so it is not divisible by $2$ or $5$.
3. Test divisibility by $3$. The sum of the digits is $2+7=9$. Because $9$ is divisible by $3$, $27$ is also divisible by $3$.

Answer: The number $27$ is composite.

Check: Since $3\times 9=27$, the factors of $27$ are $1$, $3$, $9$, and $27$. Because it has more than two factors, it cannot be prime.

Example 2: Testing a larger number

Question: Is $83$ a prime number?

Method:

1. Check if it is even. It ends in $3$, so it is odd and not divisible by $2$.
2. Check if it ends in $5$ or $0$. It ends in $3$, so it is not divisible by $5$.
3. Check divisibility by $3$. The sum of the digits is $8+3=11$, which is not divisible by $3$.
4. Check divisibility by $7$. Dividing $83÷7$ gives $11$ with a remainder of $6$.

Answer: The number $83$ is prime.

Check: The largest prime you must check is the one just before the number's square root. Since $9\times 9=81$, testing primes up to $7$ is sufficient to prove that $83$ has exactly two factors.

Example 3: Prime numbers greater than 100

Question: Is $119$ prime or composite?

Method:

1. Test small primes. It is not even, does not end in $5$, and the digits sum to $1+1+9=11$, which means it is not divisible by $3$.
2. Test divisibility by $7$. Divide $119$ by $7$.
3. Calculate $119÷7=17$. The division is exact with no remainder.

Answer: The number $119$ is composite.

Check: The factors are $1$, $7$, $17$, and $119$. Four positive factors means the number is composite.

Common Mistakes and Non-Examples

When identifying prime numbers, learners frequently fall into a few common traps regarding specific digits and rules.

• Thinking 1 is a prime number: The number $1$ only has one factor, which is itself. A prime number must have exactly two positive factors. Therefore, $1$ is a non-example; it is neither prime nor composite.
• Assuming all odd numbers are prime: Numbers like $9$, $15$, and $21$ are odd but they are composite because they are perfectly divisible by $3$.
• Believing 2 cannot be prime: The number $2$ is the only even prime number in mathematics because its only factors are $1$ and $2$. Every other even number is divisible by $2$, giving it at least three factors.
• Checking negative numbers: By definition, prime numbers are strictly positive whole numbers greater than $1$. Negative numbers are never classified as prime.

Real-World Connections

Prime numbers are the mathematical backbone of modern digital security. When you send a secure message, use a banking application, or purchase items online, the encryption algorithms securing your data rely on enormous prime numbers.

Computers create public keys by multiplying two massive prime numbers together. Because it is incredibly difficult and time-consuming for computers to work backwards and find the original prime factors of a number that is hundreds of digits long, prime numbers keep your digital information completely safe from unauthorized access.