[bananarama2]

How do you solve simultaneous modular equations?

E.g




I know guessing gives one solution: a=1,b=-10

[marmeduke]

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)

[ztibor]

(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)

Adding together the 2 eq.
34a+2b=14 (mod 26)
with a=1 (mod 26)
2b=-20 (mod 26)
b=-10 (mod 26)

[Carolus]

(Original post by ztibor)
WIth subtracting
12a =12 (mod 26)
a=1 (mod 26)

Adding together the 2 eq.
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.