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Math Help - Canonical representation?

  1. #1
    Member aldrincabrera's Avatar
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    Cool Canonical representation?

    ,.good day everyone,.,need ur help with how to prove this,.
    Let n>=2 be an integer with canonical representation n=p₁^{a₁}p₂^{a₂}p₃^{a₃}...p_{k}^{a_{k}}. an integer m>=1 is a positive divisor of n if and only if m=p₁^{b₁}p₂^{b₂}p₃^{b₃}...p_{k}^{b_{k}}, where 0<=b_i<=a_i for all 1<=i<=k.
    thnx in advance...
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Canonical representation?

    If \{p_i\} is the set of all prime numbers, we can express n,m, s positive integers as unique products

     n=\prod{p_i^{\alpha_i}} , m=\prod {p_i^{\beta_i}} , s=\prod {p_i^{\gamma_i}}

    where \alpha_i\geq 0,\beta_i\geq 0,\gamma_i\geq 0 are finitely many non zero integers.

    If \beta_i\leq \alpha_i for all i evidently m|n . If m|n there exists s positive integer such that n=ms that is, \prod p_i^{\alpha_i}=\prod p_i^{\beta_i+\gamma_i} . By the unique factorization, \alpha_i=\beta_i+\gamma_i , hence \beta_i\leq \alpha_i for all i .
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