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.

You do know that $X\equiv Z\pmod{bd}$? So how is $Y$ between them? — Rushabh Mehta, Nov 27 '19 at 17:21
@StevenStadnicki Of course I know you can do that, but the way it is phrased, it seem as though $X<Y<Z$ is given by the equivalences. — Rushabh Mehta, Nov 27 '19 at 17:32
@DonThousand Hrm; I can see where you're coming from, but I just took $X\lt Y\lt Z$ as another piece of information the user starts with, rather than something that they've derived from the equalities. — Steven Stadnicki, Nov 27 '19 at 17:39
Thanks for everyone, was =+ and =+2. Sorry if the question was too obvious. — user3066261, Nov 27 '19 at 17:47

Bill Dubuque · Answer 1 · 2019-11-27T17:35:16.927

0

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 $\,n\ge 0,\,$ $\,y\le z$

edited Nov 27 '19 at 17:35

answered Nov 27 '19 at 17:30

Bill Dubuque

272,048

I need a modular arithmetic theorem or method

1 Answers1