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Math Help - Order modulo proof help

  1. #1
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    Order modulo proof help

    If xy ≡ 1 (modulo n) st x, y, n are natural numbers, then x, y have the same order modulo n.

    I tried direct proof and contradiction, but was stuck both ways.

    I really need help on dis, thank u.
    -Frank
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  2. #2
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    Hi

    If xy ≡ 1 (modulo n) st x, y, n are natural numbers, then x, y have the same order modulo n.
    The second "n" is different form the first one, isn't it?

    xy\equiv 1 (\text{mod}n)\Rightarrow x,y \in (\mathbb{Z}_n^*,\times)

    Let k be the order of x, i.e. x^k=1 (\text{mod}n) and for any 0<l|k,\ x^l\neq 1(\text{mod}n)

    What about (xy)^k ? Let l be a positive divisor of k, what can you say about (xy)^l\ \text{if}\ y^l=1 ? Conclusion?
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