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Math Help - congruences

  1. #1
    Senior Member Sampras's Avatar
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    congruences

    Suppose you have equations like  x+x+x+x+x = 0 in  \mathbb{Z}_5 .

     \mathbb{Z}_5 = \{[0], [1], [2], [3], [4] \} . So if we pick any representative from these congruence classes they will solve the equation. So  x = 0,1,2,3,4 . But  x = 5,6,7,8,9 will also work right? Because they are representatives? But these values are "equal" to the other values.

    Also the  x is just a symbol so could we just do the following:  [x] + [x] + [x] + [x] + [x] = [5x] =[0] so that  x = [0] ?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Suppose you have equations like  x+x+x+x+x = 0 in  \mathbb{Z}_5 .

     \mathbb{Z}_5 = \{[0], [1], [2], [3], [4] \} . So if we pick any representative from these congruence classes they will solve the equation. So  x = 0,1,2,3,4 . But  x = 5,6,7,8,9 will also work right? Because they are representatives? But these values are "equal" to the other values.

    Also the  x is just a symbol so could we just do the following:  [x] + [x] + [x] + [x] + [x] = [5x] =[0] so that  x = [0] ?
    Right they all work.

    Also, [x] + ... + [x] = [5x] = [0] since 5x is always divisible by 5. Therefore, any x will solve [x] + ... + [x] = [0].
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