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  • Discrete math proof needed

    Anonymous Christian
    Sep 5, 2014
    MATH
    Comments Off on Discrete math proof needed

    Question

    It is asking for the proof for lemma X. According to the discussion in class we know that when you have something like a mod j. that translate into:

    a = jq + r where 0 <= r < j
    also r has to be the smallest number possible. I just don’t know how to go about it. The lema in question is showned below.

    Lemma X : Given j > 1 and k > 1. If a mod jk = 1 then
    a mod j = 1 and a mod k = 1.

    how can I prove that?

    Answer

    Lemma X : Given j > 1 and k > 1. If a mod jk = 1 then
    a mod j = 1 and a mod k = 1.

    Firstly,
    a mod jk = 1 means,
    a = 1 + jkh for some integer h.

    Now call kh = M and see that,
    a = 1 + jM. So, a mod j = 1. because our ‘r’ here is ‘1’ and satisfies, 0 <= 1 < j. And using the “definition” of “mod” that you gave.

    Similarly call jh = N and see that,
    a = 1 + kN. So a mod k = 1. because our ‘r’ here is ‘1’ and satisfies, 0 <= 1 < j. And using the “definition” of “mod” that you gave.

    QED.

anonymous-christian

Jonathan Ramachandran is founder at AnonymousChristian.org which serves as a Christian Blog discussing the Greater Hope of Biblical Christian in Christ Alone. He is also an IMO (International Math Coach) and a featuring Guitarist for many bands.

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