Let a, b, c be positive integers satisfying: $$ a \equiv 0 \;(mod\;3)\\ b \equiv 0 \;(mod\;5)\\ c \equiv 0 \;(mod\;7)\\ a+b \equiv 0 \;(mod\;67)\\ b+c \equiv 0 \;(mod\;17)\\ c+a \equiv 0 \;(mod\;73)\\ $$
This problem requires the smallest possible value of a + b + c.
My approach was to solve this congruence equation with an attempt to obtain some general formulae for a, b and c but it did not seem to work.
I would like to know how simultaneous equations in different congruence can be solved in general (if there is any reference for that) or whether there is any trick to handle this particular problem without considering the general problem.
Hint
Let $a+b+c=x$, then the last three congruences can be written as \begin{align*} x \equiv c \pmod{67}\\ x \equiv a \pmod{17}\\ x \equiv b \pmod{73} \end{align*} Since the modulii are relatively prime use Chinese Remainder Theorem. Of course you will need to incorporate the first three congruences ultimately, but that will be an easy task.