anonymous
  • anonymous
can I divide by 2? 8x ≡ 12 (mod 20)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
to get 4x ≡ 6 (mod 10) ??
ganeshie8
  • ganeshie8
sure you can : \(8x\equiv 12\pmod{20} \implies 20\mid (8x-12) \\~\\\implies 20\mid 2(4x-6) \implies 10\mid (4x-6)\)
ganeshie8
  • ganeshie8
you can divide by \(2\) again if you want to

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anonymous
  • anonymous
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) ??
ganeshie8
  • ganeshie8
thats right, notice that \[ab \mid ac \implies b\mid c\]
ganeshie8
  • ganeshie8
\(2*10\mid 2(4x-6) \implies 10\mid (4x-6)\)
anonymous
  • anonymous
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)
anonymous
  • anonymous
gcd(8,20)*
ganeshie8
  • ganeshie8
You want to solve \(8x \equiv 12 \pmod {20} \) the fastest way to do this is to divide \(8\) through out
ganeshie8
  • ganeshie8
\(\large 8x \equiv 12 \pmod {20}\) \(\large x \equiv \dfrac{12}{8} \pmod {\dfrac{20}{\gcd(8,20)}}\) \(\large x \equiv \dfrac{12}{8} \pmod {5}\)
ganeshie8
  • ganeshie8
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 \(x\equiv 4\pmod{5}\)
anonymous
  • anonymous
you did you get 6/4 ≡ 6/(-1)? Looks like you did 4 - 5 in the denominator?
ganeshie8
  • ganeshie8
our goal is to convert that fraction into an integer 4 is same as -1 in mod 5, so...
ganeshie8
  • ganeshie8
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..
ganeshie8
  • ganeshie8
i just want to show you that division works pretty naturally with congruences whenever the ivnverses are defined
anonymous
  • anonymous
Maybe I'll stick to guess and check for now. I will study this method further
ganeshie8
  • ganeshie8
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
anonymous
  • anonymous
thank you @ganeshie8 :')
ganeshie8
  • ganeshie8
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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