Okay I don't know what way you people have gone about solving it, I haven't done questions like this either, hence why I am posting my solution, as I'm not sure if its the correct approach.
But I solved it like this... (Not sure if this is ideal, but it gave me a solution)
7 = 7 x 1, -7 x -1
55 = 5 x 11, -5 x -11, 55 x 1, -55 x -1
Common factors are 1 and -1
Since the coefficents of m and n are so large and we are looking for integers of m and n...
|m,n| greater than and equal to 1...
from this we know that if +/- m, then -/+ n to be possible to get a end value of 1 (Basically if one is positive the other is negative).
Test m = 1
7 + 55n = 1
55n = -6
n = -6/55
Test m = -1
-7 + 55n = 1
55n = 8
n = 8/55
Test n = 1
7m + 55 = 1
7m = -54
m = -54/7
Test n = -1
7m -55 = 1
7m = 56
m = 8
Solution found... n = -1 and m = 8