## "Characteristics" of numbers

Mona is very popular in her class. She has the following characteristics:

- likes listening to music
- likes to swim
- likes horses
- likes to eat pizza
- plays on the piano

Numbers have characteristics as well. For example the number 12:

Every number is divisible by 1 and itself, so it's about as interesting as saying that Mona has birthday once a year and eyes, ears and nose.

Most important are the factors 2 and 3. They are prime numbers:

## Definition: prime number

*A prime number exactly has 2 factors: 1 and itself.*

The number 1 is not a prime number, since its only factor is 1.

The first prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19...

Try to find more!

Life is much easier if you memorize at least the first prime numbers.

__Online Exercise:__

This important exercise teaches you to recognize whether
a given number is a prime number ist.

If you have any problems with this exercise, please have a look at the **hint** and the **sample solution** first.

## Definition: factor

The numbers used for multiplication are called **factors**. The result is called **product**:

*factor × factor = product*

## Definition: prime factor

If all factors are prime numbers then they are called **prime factors**.

Any number that is not a prime number can be disassembled into its prime factors. Multiplying these prime factors results in the number.

## Formula: prime factorization

Let's try to disassemble the number **12** into its prime factors. So we are looking for prime numbers, which multiplied by each other give the result 12:

### Step 1: Try if the number 12 is dividable by the first prime number - 2:

Divide 12 by 2:

12 ÷ 2 = 6

Since the division works without remainder, we found the first prime factor: **2**

The first disassembly of the number 12 is **2 × 6 = 12**

The number 6 is not a prime number - so we disassmble this number further:

### Step 2: Try if the number 6 is dividable by the first prime number - 2:

Divide 6 by 2:

6 ÷ 2 = 3

Since the division works without remainder, we found the second prime factor: also **2**

The new disassmbly of the number 12 is **2 × 2 × 3 = 12**

Because 3 is a prime number, we disassmbled the number 12 completely into its prime factors: **2 × 2 × 3 = 12**

Another example, this time it's more complicated:

Let's disassemble the numer 980:

Try the factor 2 → 980 ÷ 2 = 490 → first prime factor: **2**

Try the factor 2 → 490 ÷ 2 = 245 → second prime factor: **2**

Try the factor 2 → 245 ÷ 2 = does not work

Try the factor 3 → 245 ÷ 3 = does not work

Try the factor 5 → 245 ÷ 5 = 49 → third prime factor: **5**

Try the factor 5 → 49 ÷ 5 = does not work

Try the factor 7 → 49 ÷ 7 = 7 → fourth prime factor: **7**

**7** is a prime number - therefore it can't be disassembled any further. Be aware that it is the fifth and final prime factor.
The disassembly is therefore as follows:

**2 × 2 × 5 × 7 × 7 = 980**

Again, a little bit more clearly arranged:

## Videos: Prime Factorization

*Prime Numbers: "The building blocks of all positive whole numbers"*

Finally, two videos about prime factorization. The first video explains the idea and terms of prime factorization. The second video explains the prime factorization using short division: