Fundamental Theorem of Arithmetic.
Each integer greater than $1$ can be written as a
product of primes, and, except for the order in which
these primes are written, this can be done in only one
way.
Lemma13.1.
If $a,$ $b,$ and $c$ are integers, with $a\mid bc$
and $(a,b)=1,$ then $a\mid c.$
Lemma13.2.
If $p$ is a prime, $a_1,\ldots,a_n$
are integers, and $p\mid a_1\cdots a_n,$
then $p\mid a_i$ for some
$i, (1\le i\le n).$
Definition.
(Page 68)
By arranging the prime factors in increasing order,
we see that each integer $n\gt1$ can be written in
the form
\[
n=p_1^{e_1}\cdots p_k^{e_k} \quad (p_1 \lt\cdots\lt p_k)
\]
where the primes $p_1,\ldots,p_k$ and the positive integers
$e_1,\ldots,e_k$ are uniquely determined by $n.$
We shall call this the
standard form for $n.$
For example, the standard form for $300$ is
$2^2 \cdot 3 \cdot 5^2.$
Alex Notes