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Unique Factorization

Theorem 31.7 For all primes p and all integers a and b, if p $\mid$ ab, then p $\mid$ a or p $\mid$ b.

Theorem 31.8 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 pi are prime, \(p_1 < p_2 < \cdots < p_r\), and the ei are positive integers.

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


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