I was solving this problem which reduces down to solving the following n variable equations :

a1x1 + a2x2 + a3x3 +.... = 1mod2
b1x1 + b2x2 + b3x3 + ... = 1mod2
..

I am aware that I have to use something on the lines of Gaussian algorithm. However simple Gauss is not giving the correct answer as it involves modular arithmetic. How do I solve such equations involving multiple mods ? Thank you :)