The following question was confusing me so any help would be appreciated!
Find the solution in integers to the equation
155x+341y+385z=1 (where 1 is the gcd of 155,341, and 385).
Okay, rewrite this equation as,
$\displaystyle 155x+341y = 1 - 385 z \ \ \ (1)$.
Now $\displaystyle \gcd(155,341)=31$ this means (1) has a solution whenever $\displaystyle 31 | (1-385z)$, i.e. $\displaystyle 385z\equiv 1 (\bmod 35)$.
But this congruence has no solution to it because $\displaystyle 385 \equiv 0$ under this modulos. Thus, this equation has no solutions.