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anonymous

  • one year ago

can I divide by 2? 8x ≡ 12 (mod 20)

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  1. anonymous
    • one year ago
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    to get 4x ≡ 6 (mod 10) ??

  2. ganeshie8
    • one year ago
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    sure you can : \(8x\equiv 12\pmod{20} \implies 20\mid (8x-12) \\~\\\implies 20\mid 2(4x-6) \implies 10\mid (4x-6)\)

  3. ganeshie8
    • one year ago
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    you can divide by \(2\) again if you want to

  4. anonymous
    • one year ago
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    So general, if ax ≡ b (mod c) and a,b,c have a common factor d and i can do (a/d)x ≡ (b/d) (mod c/d) ??

  5. ganeshie8
    • one year ago
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    thats right, notice that \[ab \mid ac \implies b\mid c\]

  6. ganeshie8
    • one year ago
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    \(2*10\mid 2(4x-6) \implies 10\mid (4x-6)\)

  7. anonymous
    • one year ago
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    ah I see. Is there a way to find the solution without trial and error initially? I know that if n is a solution then all solutions is x = n + t*20/gcd(8/20)

  8. anonymous
    • one year ago
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    gcd(8,20)*

  9. ganeshie8
    • one year ago
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    You want to solve \(8x \equiv 12 \pmod {20} \) the fastest way to do this is to divide \(8\) through out

  10. ganeshie8
    • one year ago
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    \(\large 8x \equiv 12 \pmod {20}\) \(\large x \equiv \dfrac{12}{8} \pmod {\dfrac{20}{\gcd(8,20)}}\) \(\large x \equiv \dfrac{12}{8} \pmod {5}\)

  11. ganeshie8
    • one year ago
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    Now look at \(\dfrac{12}{8}\) in mod 5 : \[\dfrac{12}{8} = \dfrac{6}{4}\equiv \dfrac{6}{-1} \equiv -6\equiv 4\] therefore the solution is \(x\equiv 4\pmod{5}\)

  12. anonymous
    • one year ago
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    you did you get 6/4 ≡ 6/(-1)? Looks like you did 4 - 5 in the denominator?

  13. ganeshie8
    • one year ago
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    our goal is to convert that fraction into an integer 4 is same as -1 in mod 5, so...

  14. ganeshie8
    • one year ago
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    you don't need to do it that way if it doesn't look intuitive... you can solve it the long way using reverse euclid gcd algorithm or by some other means..

  15. ganeshie8
    • one year ago
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    i just want to show you that division works pretty naturally with congruences whenever the ivnverses are defined

  16. anonymous
    • one year ago
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    Maybe I'll stick to guess and check for now. I will study this method further

  17. ganeshie8
    • one year ago
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    this is not a big method as such as you can see we're treating it as a regular algebraic equation and dividing stuff both sides

  18. anonymous
    • one year ago
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    thank you @ganeshie8 :')

  19. ganeshie8
    • one year ago
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    np, with the orthodox method you will be solving \(2x\equiv 3\pmod{5}\) by using reverse euclid division algorithm, step by step, I think

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