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Thread: Need help with 3 homework quesions

  1. #1
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    Need help with 3 homework quesions

    Here are my 3 problems:

    Prove or disprove that (n^2)-1 is composite whenever n is a positive integer greater than 1
    Prove that (a mod m)(b mod m) mod m = ab mod m for all integers a and b whenever m is a positive integer.
    Prove or disprove that a mod m + b mod m = (a + b) mod m for all integers a and b whenever m is a positive integer.

    Help to Any of these would be appreciated..

    thanks
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  2. #2
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    Code:
    n^2-1
    Composite
    Notice that, $\displaystyle n^2-1=(n-1)(n+1)$ can be factored, thus it is not prime i.e. composite.
    Unless, the factors are trivial, meaning that $\displaystyle n-1=1$ thus, $\displaystyle n=2$ is the only prime. Thus, $\displaystyle 2^2-1=3$ is the only prime obtained by this function.
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  3. #3
    Grand Panjandrum
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    Prove that (a mod m)(b mod m) mod m = ab mod m for all integers a and b whenever m is a positive integer.
    Let

    $\displaystyle
    a=k_1\times m +c_1,\ c_1 \in \{0, \dots m-1\}
    $,

    and:

    $\displaystyle
    b=k_2\times m +c_2,\ c_2 \in \{0, \dots m-1\}
    $.

    Then:

    $\displaystyle
    (a\mod m)(b\mod m) \equiv c_1c_2 \mod m
    $

    But

    $\displaystyle
    ab \mod m=\{k_1k_2 m^2 + (k_1c_2+c1k_2)m+c_1c_2\}\mod m$$\displaystyle =c_1c_2 \mod m
    $.

    Hence:

    $\displaystyle
    (a \mod m)(b \mod m) \mod m = ab \mod m
    $

    RonL
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