x congruent to 4 (mod 11)

x congruent to 3 (mod 17)

M = 11 * 17 = 187

so M1 = 11, M2 = 17

i'm trying to find the inverse of M1 and M2 but am unsure how because M1M1' congruent to 1 (mod 11) but M1M1' is a factor of 11 so it will always have a remainder of 0.

can this be proven?