Prime factor facts for kids

A prime factor is a special kind of number that is both a factor of another number and a prime number itself. Think of them as the unique, unbreakable building blocks that make up all other whole numbers!

What is a Factor?

Before we talk about prime factors, let's make sure we understand what a "factor" is.

Definition: A factor of a number is a whole number that divides evenly into that number, leaving no remainder. Factors are like partners in multiplication! When you multiply two factors together, you get the original number.

Example Time! Let's take the number 12. What numbers can divide into 12 evenly?

1 can divide into 12 (1 x 12 = 12) 2 can divide into 12 (2 x 6 = 12) 3 can divide into 12 (3 x 4 = 12) 4 can divide into 12 (4 x 3 = 12) 6 can divide into 12 (6 x 2 = 12) 12 can divide into 12 (12 x 1 = 12)

So, the factors of 12 are 1, 2, 3, 4, 6, and 12. See how they all fit perfectly into 12 without any leftovers?

What is a Prime Number?

Now that we know about factors, let's meet the "prime" part of prime factors!

Definition: A prime number is a whole number greater than 1 that has only two factors: 1 and itself.

Think of prime numbers as the "loners" or "independent" numbers. They don't have many friends when it comes to factors.

Fun Facts:

• The number 1 is NOT a prime number. It only has one factor (itself).
• The number 2 is the smallest prime number, and it's the only even prime number! All other even numbers can be divided by 2, so they have more than two factors.

Examples of Prime Numbers:

2: Its only factors are 1 and 2.
3: Its only factors are 1 and 3.
5: Its only factors are 1 and 5.
7: Its only factors are 1 and 7.
11: Its only factors are 1 and 11.
13: Its only factors are 1 and 13.
...and many, many more!

Numbers that are not prime (and are greater than 1) are called composite numbers. Composite numbers have more than two factors. For example, 4 is a composite number because its factors are 1, 2, and 4.

Prime Factors

When we combine our understanding of "factors" and "prime numbers," we get prime factors.

Definition: A prime factor of a number is a factor of that number that is also a prime number.

Example: Let's go back to our number 12. Its factors are: 1, 2, 3, 4, 6, 12. Now, let's look at this list and pick out the ones that are also prime numbers:

2 is a prime number (factors are 1 and 2).
3 is a prime number (factors are 1 and 3).
1, 4, 6, and 12 are NOT prime numbers.

So, the prime factors of 12 are 2 and 3.

When we talk about prime factorization, we're talking about breaking a number down completely into all of its prime factors, even if they repeat.

For 12, we can write it as a product of its prime factors: 12 = 2 x 2 x 3

Notice that the number 2 appears twice! This is important. Every number has a unique set of prime factors.

How to Find Prime Factors (Prime Factorization)

There are two super fun ways to find the prime factors of a number: the Factor Tree method and the Division method.

The Factor Tree

This method is like drawing a tree where the branches split into factors until you only have prime numbers at the ends of the branches.

Let's find the prime factors of 36 using a Factor Tree:

Start with the number at the top:
36
/ \

Find any two factors that multiply to give 36 (don't use 1). Let's pick 6 and 6:
36
/ \
6 6

Now, look at each of those factors. Are they prime? No, 6 is not prime (factors are 1, 2, 3, 6). So, we break them down further:
36
/ \
6 6
/ \ / \
2 3 2 3

Now, look at the numbers at the very bottom of your branches: 2, 3, 2, 3. Are they all prime numbers? Yes! 2 is prime, and 3 is prime.

You've reached the end of your tree! The prime factors of 36 are 2, 2, 3, and 3. We can write this as: 36 = 2 x 2 x 3 x 3.

The Division Method

This method involves repeatedly dividing the number by the smallest possible prime number until you can't divide anymore.

Let's find the prime factors of 36 using the Division Method:

Start with the number 36. What's the smallest prime number that can divide 36 evenly? It's 2!
2 | 36
----

Divide 36 by 2. You get 18. Write 18 below 36:
2 | 36
----
18

Now, take 18. What's the smallest prime number that can divide 18 evenly? It's 2 again!
2 | 36
----
2 | 18
----

Divide 18 by 2. You get 9. Write 9 below 18:
2 | 36
----
2 | 18
----
9

Now, take 9. What's the smallest prime number that can divide 9 evenly? It's not 2. The next smallest prime is 3!
2 | 36
----
2 | 18
----
3 | 9
----

Divide 9 by 3. You get 3. Write 3 below 9:
2 | 36
----
2 | 18
----
3 | 9
----
3

Now, take 3. What's the smallest prime number that can divide 3 evenly? It's 3 itself!
2 | 36
----
2 | 18
----
3 | 9
----
3 | 3
----
1

You stop when you reach 1 at the bottom. The prime factors are all the numbers on the left side of your division: 2, 2, 3, 3.

Again, the prime factors of 36 are 2, 2, 3, and 3. Both methods give you the same answer!

Why Are Prime Factors Important?

Prime factors might seem like a fun math puzzle, but they are actually super useful in many ways.

As we mentioned, every whole number (greater than 1) has one and only one set of prime factors. This is a very famous idea in math called the Fundamental Theorem of Arithmetic. It means prime numbers are truly the basic "atoms" of numbers!

When you need to simplify a fraction (like 12/36), finding the prime factors of the top and bottom numbers can help you find the biggest number you can divide both by (called the Greatest Common Factor, or GCF). This makes simplifying fractions much easier!

Prime factors also help you find the smallest number that two or more numbers can both divide into (called the Least Common Multiple, or LCM). This is useful when you're adding or subtracting fractions with different denominators.

Believe it or not, prime numbers and prime factors are used in making sure your online messages and information are safe! When you send a message or buy something online, special codes (called cryptography) are used to protect your data. These codes often rely on multiplying two very large prime numbers together. It's incredibly hard for computers to break these huge numbers back down into their prime factors, which keeps your information secure!

Fun Facts About Prime Numbers

• Mathematicians are always looking for bigger and bigger prime numbers. The largest known prime number today has millions and millions of digits!
• Some prime numbers like to hang out in pairs. A "twin prime" is a pair of prime numbers that are separated by just one even number. For example, (3, 5), (5, 7), (11, 13), and (17, 19) are all twin primes. Mathematicians are still trying to figure out if there are infinitely many twin primes.