Next: Up: Previous:

Unique Factorization

Theorem 33.7 For all primes p and all integers a and b, if p $\mid$ ab, then p $\mid$ a or p $\mid$ b.

Theorem 33.8 A composite integer a can be written in exactly one way as a product of the form $a \;=\; p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r}$ , where the p_i are prime, $p_1 < p_2 < \cdots < p_r$ , and the e_i are positive integers.

Examples: $675 \;=\; 3^3 \;*\; 5^2$
$1350 \;=\; 2 \;*\; 3^3 \;*\; 5^2$
$255 \;=\; 3 \;*\; 5 \;*\; 17$

Next: Up: Previous: