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# Simultaneous modular equations

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1. Simultaneous modular equations
How do you solve simultaneous modular equations?

E.g

I know guessing gives one solution: a=1,b=-10
Last edited by bananarama2; 04-06-2012 at 20:19.
2. Re: Simultaneous modular equations
rearrange to get 2 equations for b mod26. you know b is congruent to itself, so you get an equation for a mod26. solve this using standard results (assuming you do know standard results to solve this)
3. Re: Simultaneous modular equations
(Original post by wcp100)
How do you solve simultaneous modular equations?

E.g

I know guessing gives one solution: a=1,b=-10
WIth subtracting
12a =12 (mod 26)
a=1 (mod 26)

34a+2b=14 (mod 26)
with a=1 (mod 26)
2b=-20 (mod 26)
b=-10 (mod 26)
4. Re: Simultaneous modular equations
(Original post by ztibor)
WIth subtracting
12a =12 (mod 26)
a=1 (mod 26)

34a+2b=14 (mod 26)
with a=1 (mod 26)
2b=-20 (mod 26)
b=-10 (mod 26)
This misses a solution. Since

there isn't a unique solution to

You can rearrange to get

which, if you then notice that 12=6*2, can be written as

which is true for

or

Then you can use the two values of a to find the corresponding values of b. It's simpler to just substitute the value of a into one of the equations than to mess around with summing them as ztibor has done.
Last edited by Carolus; 11-06-2012 at 04:26.

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Last updated: June 11, 2012
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