Resolve a system using Chinese remainder theorem

Question

I have the following system: $x \equiv 3 \mod 26$ and $x \equiv 7 \mod 41$

How can we solve it using Chinese remainder theorem. Any guidelines ? Many thanks

@MarwanB you probably got (-1) because you didn't include in the question any information about what you tried, and also adding the theorem in the question can be helpful — Holo, Dec 25 '17 at 12:24
I guess the only reasonable answer is by applying the Chinese remainder theorem. Have you already managed to solve similar problems? — Jack D'Aurizio, Dec 25 '17 at 14:32

score 2 · Answer 1 · answered Dec 25 '17 at 13:14

The general method consists in finding first a Bézout's relation between $26$ and $41$, $\;26\,u+41\,v=1,\; u,v \in\mathbf Z$ with the extended Euclidean algorithm.

Then, you have a formula for the solutions: $$x\equiv7\cdot 26\,u+3\cdot41\,v\mod 26\cdot41.$$

You should find $\; x\equiv -75\mod 1066$.

score 1 · Answer 2 · answered Dec 25 '17 at 12:05

1

you can write $$x=3+26m,x=7+41n$$ so we get $$26m-41n=4$$ with $$m,n\in \mathbb{Z}$$ solving this Diophantine equation we get $$m=348+41k,n=24+26k$$

answered Dec 25 '17 at 12:05

Dr. Sonnhard Graubner

97,058

Is that CRT?.... – nonuser Dec 25 '17 at 12:06
we do not need that, why do you ask me knowing the answer? – Dr. Sonnhard Graubner Dec 25 '17 at 12:07
Actually I need that because I am asked to use CRT. How did you solve the Diophantine? did you use a solver ? Thanks – Conjecture Dec 25 '17 at 12:14
i solved it with remainder calculus – Dr. Sonnhard Graubner Dec 25 '17 at 12:15

Resolve a system using Chinese remainder theorem

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