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Status: Tags: #archivedCards/macm101/numbertheory Links: Number Theory

Composite Numbers

Proof: Every composite number has a prime divisor ?

• Let S be the set of all composite numbers that do not have a prime divisor
• Since S ⊆ N, by the Well-Ordering Principle, it has a least element r.
• As r is not prime, it has a divisor, therefore, r = uv for some positive integers u and v. u < r and v < r.
• Therefore u ∉ S, and u has a prime divisor p.
• Since p | u and u | r, we conclude that p | r, a contradiction.

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list from Composite Numbers AND !outgoing(Composite Numbers)

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Created:: 2021-11-25 15:57