# What is the product of prime numbers

### Prime factorization

You already know prime numbers: They are the numbers that have exactly two factors. Prime numbers are only divisible by 1 and by themselves.

The number 1 is not a prime number. It has only one factor, the 1.

These are all prime numbers that are less than 100:

$$2$$ $$3$$ $$5$$ $$7$$

$$11$$ $$13$$ $$17$$ $$19$$

$$23$$ $$29$$

$$31$$ $$37$$

$$41$$ $$43$$ $$47$$

$$53$$ $$59$$

$$61$$ $$67$$

$$71$$ $$73$$ $$79$$

$$83$$ $$89$$

$$97$$

You can write all natural numbers as the product of prime numbers. It doesn't sound that exciting at first, but it can be useful for arithmetic.

**Examples:**

The numbers 15 and 66 with their prime factorization:

$$15=3*5$$

$$66=2*3*11$$

To the right of $$ = $$ there are only prime numbers: 3 and 5 for the 15 or 2 and 3 and 11 for the 66.

Any natural number that is not itself a prime number can be broken down into a product of prime numbers.

### Find the prime factorization yourself

How do you find the prime factorization of a number?

Exercise: Write 108 as the product of prime numbers.

Find a number that 108 is divisible by. 108 is an even number, so it's divisible by 2.

$$108 = 2*54$$

54 is also straight. So divide 54 by 2.

$$108 = 2*2*27$$

The number 27 is divisible by 3. Divide 27 by 3.

$$108 = 2*2*3*9$$

The number 9 is divisible by 3

$$108=2 * 2 * 3 * 3 * 3$$

You cannot break down the factors on the right. These are all prime numbers now.

Write down the prime factorization even more briefly: with the power notation.

$$108 = 2^2* 3^3$$

You look good at a number

- whether it is divisible by 2: last digit is even
- whether it is divisible by 5: last digit 0 or 5
- whether it is divisible by 10: last digit 0
- whether it is divisible by 3: checksum by 3

If a factor occurs several times, use the power notation.

Example:

$$100 = 2^2 * 5^2$$.

You know what?

$$4^3 = 4 * 4 * 4$$

└──┬─┘

$$ 3 $$ - times the factor $$ 4 $$

Powers always look like this:

Read: 4 to the power of 3

### Different calculation methods

There are different ways of calculating to find the prime factorization.

At 108 you can only calculate through 4. (8 is divisible by 4 and 100 too.)

$$108=4*27$$

4 is 2 times 2.

$$108=2*2*27$$

27 is divisible by 3.

$$108=2*2*3*9$$

9 is also divisible by 3.

$$108=2*2*3*3*3$$

With potencies:

$$108=2^2*3^3$$

There are different ways of calculating to find the prime factorization. They all lead to the same result. Because you can swap factors in a product (commutative law).

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### Another example

Exercise: Write 920 as the product of prime numbers.

920 ends in 0. Divide by 10 first.

$$920= 10*92$$

You can write 10 as 2 $$ * $$ 5.

$$920 = 2*5*92$$

92 is an even number. Calculate by 2.

$$920 = 2*5*2*46$$

46 is an even number, i.e. divided by 2.

$$920 = 2*5*2*2*23$$

23 is a prime number. You cannot disassemble any further.

It still looks nicer in this order:

$$920 = 2*2*2*5*23$$

And with potencies:

$$920= 2^3*5*23$$

When you prime a number, divide until there are only prime numbers in the product.

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