The theory of numbers, by Robert D. Carmichael ...

16

THEORY OF NUMBERS

Evidently the process may be continued until one side of the
equation is reduced to i. The other side must also be reduced
to i at the same time. Hence it follows that the two decompositions of m are in fact identical.
This completes the proof of the theorem.
The result which we have thus demonstrated is easily the
most important theorem in the theory of integers. It can
also be stated in a different form more convenient for some
purposes:
II. Every non-unit positive integer m can be represented in
one and in only one way in the form
m= plp2a2... pn
where pl, p2,..., pn are different primes and ai, a2,..
an are positive integers.
This comes immediately from the preceding representation
of m in the form m=pip2... pr by combining into a power
of pi all the primes which are equal to pi.
COROLLARY I. If a and b are relatively prime intzegers
and c is divisible by both a and b, then c is divisible by ab.
COROLLARY 2. If a and b are each prime to c then ab is
prime to c.
COROLLARY 3. If a is prime to c and ab is divisible by c,
then b is divisible by c.
~ 8. THE DIVISORS OF AN INTEGER
The following theorem is an immediate corollary of the
results in the preceding section:
I. All the divisors of m,
=plalp22.. p n,
are of the form
pl p22... Pn op < < as;

and every such number is a divisor of m.