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anonymous
 one year ago
can I divide by 2? 8x ≡ 12 (mod 20)
anonymous
 one year ago
can I divide by 2? 8x ≡ 12 (mod 20)

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0to get 4x ≡ 6 (mod 10) ??

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2sure you can : \(8x\equiv 12\pmod{20} \implies 20\mid (8x12) \\~\\\implies 20\mid 2(4x6) \implies 10\mid (4x6)\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2you can divide by \(2\) again if you want to

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So 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
 one year ago
Best ResponseYou've already chosen the best response.2thats right, notice that \[ab \mid ac \implies b\mid c\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\(2*10\mid 2(4x6) \implies 10\mid (4x6)\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ah 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)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2You want to solve \(8x \equiv 12 \pmod {20} \) the fastest way to do this is to divide \(8\) through out

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\(\large 8x \equiv 12 \pmod {20}\) \(\large x \equiv \dfrac{12}{8} \pmod {\dfrac{20}{\gcd(8,20)}}\) \(\large x \equiv \dfrac{12}{8} \pmod {5}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2Now 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
 one year ago
Best ResponseYou've already chosen the best response.0you did you get 6/4 ≡ 6/(1)? Looks like you did 4  5 in the denominator?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2our goal is to convert that fraction into an integer 4 is same as 1 in mod 5, so...

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2you 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
 one year ago
Best ResponseYou've already chosen the best response.2i just want to show you that division works pretty naturally with congruences whenever the ivnverses are defined

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Maybe I'll stick to guess and check for now. I will study this method further

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2this 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
 one year ago
Best ResponseYou've already chosen the best response.0thank you @ganeshie8 :')

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2np, 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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