audio read-through Prime Numbers

The numbers $\,2, 3, 4,\, \ldots\,$ can be expressed as products in a very natural way! Just keep ‘breaking them down’ into smaller and smaller factors until you can't get the ‘pieces’ any smaller.

For example:

$360$$=$$36$$\cdot$$10$
 $=$$6$$\cdot$$6$$\cdot$$2$$\cdot$$5$
 $=$$2$$\cdot$$3$$\cdot$$2$$\cdot$$3$$\cdot$$2$$\cdot$$5$

OR

$360$$=$$4$$\cdot$$90$
 $=$$2$$\cdot$$2$$\cdot$$9$$\cdot$$10$
 $=$$2$$\cdot$$2$$\cdot$$3$$\cdot$$3$$\cdot$$2$$\cdot$$5$

OR

$360$$=$$6$$\cdot$$60$
 $=$$2$$\cdot$$3$$\cdot$$6$$\cdot$$10$
 $=$$2$$\cdot$$3$$\cdot$$2$$\cdot$$3$$\cdot$$2$$\cdot$$5$

No matter how the number is ‘broken down’, you'll always get to the same place, except for possibly different orderings of the factors.

In the example above, you always get: $$360 = 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5$$ Three factors of $\,2\,$, two factors of $\,3\,$, and one factor of $\,5\,$.

These smallest ‘pieces’ (like $\,2\,$, $\,3\,$ and $\,5\,$ above) are, in a very real way, basic ‘building blocks’ for numbers being represented as products. They're very, very, very important! So, you shouldn't be surprised that these ‘multiplicative building blocks’ are given a special name:

DEFINITION prime numbers
A counting number greater than $\,1\,$ is called prime if the only numbers that go into it evenly are itself and $\,1\,$.

Notes on the Definition

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