How to solve the following 2 linear congruences x≡2(mod7), x≡5(mod11)

Chasity Kane

Chasity Kane

Answered question


How to solve the following 2 linear congruences?
x 2 ( mod 7 )
x 5 ( mod 11 )

Answer & Explanation



Beginner2022-10-07Added 8 answers

Setting x = 2 + 7 n = 5 + 11 m, you get that 7 n 11 m = 3. In fact, since 7 and 11 are coprime you can do even better - there exist integers s , t such that 7 s + 11 t = 1. This is called Bezout's identity and s and t can be found with the extended Euclidean algorithm, but here you can probably just guess and check a solution. Of course once you find s you can triple it to get n.
In general, the existence and uniqueness of a solution is given by the Chinese remainder theorem.

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