# 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.