Prime factorization facts for kids

Prime factorization is a cool way to break down any number into its smallest building blocks. Imagine you have a big LEGO model, and you want to see all the individual LEGO bricks it's made of. Prime factorization does something similar for numbers! It helps us find the special numbers called prime numbers that multiply together to make the original number.

What is Prime Factorization?

Prime factorization means finding all the prime numbers that, when multiplied together, give you the original number. Think of it like finding the "ingredients" of a number, where all the ingredients are prime numbers.

What are Prime Numbers?

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

• For example, 2 is a prime number because its only factors are 1 and 2.
• 3 is also a prime number (factors: 1 and 3).
• 5 is a prime number (factors: 1 and 5).
• Numbers like 4 are not prime because they have more than two factors (1, 2, and 4). These are called composite numbers.

How to Find Prime Factors

To find the prime factors of a number, you keep dividing it by the smallest possible prime numbers until all the numbers you have left are prime.

Here's how you do it:

• Start with the smallest prime number, which is 2.
• See if your number can be divided by 2 without any remainder.
• If it can, divide it and write down 2 as one of your prime factors.
• Keep dividing the new number by 2 until it can't be divided by 2 anymore.
• Then, move to the next smallest prime number, which is 3.
• Repeat the process: divide by 3 as many times as possible.
• Continue with the next prime numbers (5, 7, 11, and so on) until the number you are dividing becomes a prime number itself.

Example: Prime Factorization of 1729

Let's find the prime factors of the number 1729.

1. Step 1: Try dividing by small prime numbers.

• Is 1729 divisible by 2? No, because it's an odd number.
• Is 1729 divisible by 3? To check, add its digits: 1+7+2+9 = 19. Since 19 is not divisible by 3, 1729 is not divisible by 3.
• Is 1729 divisible by 5? No, because it doesn't end in 0 or 5.
• Is 1729 divisible by 7? Let's try: 1729 ÷ 7 = 247. Yes, it is!

So, we can write: 1729 = 7 × 247

1. Step 2: Factor the remaining number.

Now we need to find the prime factors of 247.

• Is 247 divisible by 7? No (247 ÷ 7 is about 35.28).
• Let's try the next prime number, 11. Is 247 divisible by 11? No (247 ÷ 11 is about 22.45).
• Let's try the next prime number, 13. Is 247 divisible by 13? Yes! 247 ÷ 13 = 19.

So, we can now write: 1729 = 7 × 13 × 19

1. Step 3: Check if all factors are prime.

Look at the numbers we have: 7, 13, and 19.

• 7 is a prime number.
• 13 is a prime number.
• 19 is a prime number.

Since all the factors (7, 13, and 19) are prime numbers, we have finished the prime factorization!

The prime factorization of 1729 is 7 × 13 × 19.

Why is Prime Factorization Useful?

Prime factorization is a basic idea in number theory, which is a branch of mathematics. It's used in many areas, including:

• Finding the greatest common divisor (GCD) and least common multiple (LCM) of numbers.
• Cryptography: This is the science of secure communication. Many modern encryption methods, like those used to protect your online messages and banking, rely on the difficulty of factoring very large numbers into their prime factors. This makes it hard for unauthorized people to break codes.

See also

In Spanish: Factorización de enteros para niños