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Math Help - 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, 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 n-1=1 thus, n=2 is the only prime. Thus, 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

    <br />
a=k_1\times m +c_1,\ c_1 \in \{0, \dots m-1\}<br />
,

    and:

    <br />
b=k_2\times m +c_2,\ c_2 \in \{0, \dots m-1\}<br />
.

    Then:

    <br />
(a\mod m)(b\mod m) \equiv c_1c_2 \mod m<br />

    But

    <br />
ab \mod m=\{k_1k_2 m^2 + (k_1c_2+c1k_2)m+c_1c_2\}\mod m =c_1c_2 \mod m<br />
.

    Hence:

    <br />
(a \mod m)(b \mod m) \mod m = ab \mod m<br />

    RonL
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