Chapter 7: Factorization and fundamental theorem of arithmetic (2)


Prime divisibility property: assumed primepDivisible producta_1,a_2,…a_r, thenpto be divisible bya_1,a_2,…a_rAt least one factor in.

Proof: ifpto be divisible bya_1, the proof is complete. Otherwise, the assertion is applied to the producta_1(a_2a_3…a_r)It is concluded thatpMust dividea_2a_3…a_rConclusion. In other words, applicationsa=a_1Andb=a_2a_3…a_rClaims of. We knowp|abSo ifp\nmid a, the assertion indicates thatpMust divideb

Now knownpto be divisible bya_2a_3…a_r, ifpto be divisible bya_2, then it proves to be complete. Otherwise, the assertion is applied to the producta_2(a_3…a_r)obtainpMust dividea_3…a_rConclusion. If we continue this process, we will eventually getpDivide aa_i

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