## "Characteristics" of numbers

What makes Mona special?

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: