I have been struggling with this problem for a long time.
$$ 19^9 \mod 1189 = 1113\\ 29^9 \mod 1189 = 522\\ 39^9 \mod 1189 = 308\\ x^9 \mod 1189 = 377 $$
the answer is $87$. It can be easily found by factorization (the fact that $1189=29 \times 41$ and then we can use Chinese remainder theorem). However, I would like to find the answer by hand. Is it possible?
Interesting fact is $19 + 29 + 39 = 87$ which is the answer, and $1113 + 522 + 308 = 754$ which is $377\times 2$.
I summed up all the equations and have the following:
$$m^9 = 1189k + 377$$
And from here, I cannot get to anywhere else.
