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Math Help - Solving a congruence equation.

  1. #1
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    Solving a congruence equation.

    i have an equation say 22=(5x+8)mod26.how do i find x from this??please help me...
    Last edited by mr fantastic; December 30th 2009 at 07:45 PM. Reason: Changed post title
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  2. #2
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    Quote Originally Posted by arun View Post
    i have an equation say 22=(5x+8)mod26.how do i find x from this??please help me...

    You can solve this since 5 is invertible \pmod {26} . The following is done using arithmetic \pmod {26} :

    5x+8=22\Longrightarrow 5x=14\Longrightarrow x=\frac{14}{5}=14\cdot 21=294=8

    Tonio
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  3. #3
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    A slight variation on that: we want to solve 5x= 14 (mod 26). That is, 5x= 14+ 26y for some integer y. That is the same as the Diophantine equation 5x- 26y= 14. 5 divides into 26 5 times with remainder 1. That is, 26= 5(5)+ 1 so 5(-5)- 26(-1)= 1. Multiply that by 14: 5(-70)- 26(-14)= 14. That tells us that x= -70 is a solution. Since we want a number between 0 and 25 that is equivalent to -70, add 3(26)= 78: x= 78- 70= 8 (mod 26).
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