I need a modular arithmetic theorem or method

Question

I only need a theorem or method to solve the problem (if exists).

With this data and having the real value of all the data marked as letter EXCEPT "Y":

X ≡ a (mod b)
X ≡ c (mod d)
Y ≡ a (mod b)
Y ≡ c (mod d)
Z ≡ a (mod b)
Z ≡ c (mod d)

I need to find "Y". I know that "Y" is a number between X and Z. "B" and "D" are prime numbers. The numbers are very big, so I need a procedure to simplify the problem.

Thanks.

Answer 1

Hint $\ x,y,z\,$ are all solutions of the same system of congruences, which, by CRT, if solvable has solutions unique $\!\bmod \ell = {\rm lcm}(b,d),\,$ so if $\,x\le y\le z\,$ then $\, y = x + n\ell\,$ for all $\,n\ge 0,\,$ $\,y\le z$