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Thread: congruences

  1. #1
    Senior Member Sampras's Avatar
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    congruences

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

    $\displaystyle \mathbb{Z}_5 = \{[0], [1], [2], [3], [4] \} $. So if we pick any representative from these congruence classes they will solve the equation. So $\displaystyle x = 0,1,2,3,4 $. But $\displaystyle 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 $\displaystyle x $ is just a symbol so could we just do the following: $\displaystyle [x] + [x] + [x] + [x] + [x] = [5x] =[0] $ so that $\displaystyle x = [0] $?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Suppose you have equations like $\displaystyle x+x+x+x+x = 0 $ in $\displaystyle \mathbb{Z}_5 $.

    $\displaystyle \mathbb{Z}_5 = \{[0], [1], [2], [3], [4] \} $. So if we pick any representative from these congruence classes they will solve the equation. So $\displaystyle x = 0,1,2,3,4 $. But $\displaystyle 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 $\displaystyle x $ is just a symbol so could we just do the following: $\displaystyle [x] + [x] + [x] + [x] + [x] = [5x] =[0] $ so that $\displaystyle x = [0] $?
    Right they all work.

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