Prime numbers

Understanding prime numbers

A mini lesson for 7th grade math students about prime numbers. What is a prime number? What are the various rules and patterns regarding prime numbers? Let's explore further.

What is a prime number?

A prime number is a number greater than 1 that has only two factors: 1 and itself. This means the only way to multiply two whole numbers to get a prime number is by multiplying 1 and the number itself.

Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23...

Non-prime numbers: 4, 6, 8, 9, 10 (because they can be divided evenly by numbers other than 1 and themselves)

Key rules of prime numbers

The number 2:

The number 2 is the only even prime number. Any other even number can be divided by 2, which means it has more than two factors and isn't prime.

All other even numbers are not prime:

Any number that ends in 0, 2, 4, 6, or 8 is even, and since it’s divisible by 2, it can’t be prime (except for 2 itself).

1 is not a prime number:

A prime number must have exactly two factors. Since 1 only has one factor (itself), it is not considered prime.

Patterns and tricks for finding prime numbers

Divisibility test:

For small numbers, you can check if a number is prime by testing if it can be divided by any prime number smaller than itself (like 2, 3, 5, 7).

Prime numbers get rarer:

As numbers get bigger, prime numbers become less frequent. This means the larger the number, the harder it is to find prime numbers.

Prime numbers cannot end in 0, 2, 4, 6, or 8 (except for 2):

Any number that ends in an even digit is not prime, except for the number 2.

The Sieve of Eratosthenes:

A method to find prime numbers by "sieving" out multiples of primes:

• List all numbers from 2 onwards.
• Cross out all multiples of 2 (like 4, 6, 8...).
• Then cross out all multiples of 3 (like 6, 9, 12...).
• Repeat this process with the next smallest uncrossed number (like 5, then 7, and so on).

Fun fact: Infinite prime numbers

There are infinitely many prime numbers. No matter how big you go, there’s always another prime number to be found!

Practice problem:

Is the number 29 a prime number?
Solution: Test if it’s divisible by smaller prime numbers (2, 3, 5). Since none of these divide evenly into 29, it is prime!

Summary

• A prime number has only two factors: 1 and itself.
• 2 is the only even prime number.
• Prime numbers get rarer as numbers get larger.
• Use patterns and divisibility rules to help find primes!

This foundation will help you explore more advanced number theory and problem-solving in math!