For example, is a prime and it divides the number
. We see that . This is enough. Conversely, as does not divide and does not divide , does not divide
.

Theorem 6.5.5
Every natural number has a unique expression as the product of primes.

Proof.
The existence part is Theorem 5.2. Now suppose
with the 's and 's
primes. Then
, so for some .
By renumbering (if necessary) we can assume that . Now repeat
with
and
, which we know must be equal.

There are perfectly nice algebraic systems where the decomposition
into primes is not unique, for instance
, where
and and
are each ``prime''. Or alternatively,
all even numbers, where ``prime'' means
``not divisible by ''. These number systems are studied in Number Theory and in Abstract Algebra.
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2002-06-10