What we want to prove: We can write every natural number (counting number, e.g. 3, 4, 5,
6…) greater than 1 as a product of prime numbers.
For example,
. 3 and 5 are prime numbers (they are only divisible by 1 and
themselves). We want to show that every number greater than 1 can be written as the product of
prime numbers.
Proof:
 For the sake of argument, let’s assume we’re wrong, and there is at least one number that
cannot be written as a product of prime numbers. (Thus begins the classic math trick of
proof by contradiction!)
 Let’s look at the smallest number for which this is true. We’ll call it n. (We know there is
a smallest number by the Well-Ordering Principle…or by common sense.)
 We know that n is not a prime number, since we know that prime numbers are easily
expressed as a product of primes (or, really, as a product of one prime: the number itself)
 So n is a composite number. This means that n has some factors. We’ll call two of those
factors a and b such that
.
 But, since a and b are smaller than n, they must be able to be written as products of
primes. (Remember, for our argument, n was the smallest number that couldn’t be written
as a product of primes.)
 So we could write
where all those ps are prime numbers and we
could write
where all those qs are also prime numbers. (For
those of you who don’t like symbols, imagine that a and b are numbers like 15, above,
where we can write it as 3*5 or like 24 where we can write it as 2*2*2*3. All those ps
and qs just represent real numbers. The only difference is that we don’t know exactly
what those numbers are. You get the idea.)
 But this means that we could write
.
 But then, we’ve just shown that we can write n as a product of primes! So our original
“for the sake of argument” proposition was false.
 This means we can write every natural number greater than 1 as a product of prime
numbers.