
Additional topics in this section: 
The Chinese Remainder Theorem is a statement about simultaneous congruences:
Suppose n_{1}, n_{2}, n_{3}, ..., n_{k} are pairwise coprime integers. Then for any given integers a_{1}, a_{2}, a_{3}, ..., a_{k}, there exists an integer, x, which is the solution to the system of equations,
x = a_{1} (mod n_{1}),
x = a_{2} (mod n_{2}),
x = a_{3} (mod n_{3}),
. . .
x = a_{k} (mod n_{k})
And if x is a solution to the system of equations, then x + kN, where N = n_{1}n_{2}n_{3}...n_{k} is also a solution.
Let N = n_{1}n_{2}n_{3}...n_{k}. Now,
for each i, consider N/n_{i}, which is the product of all the n's except
n_{i}.
Find the reciprocal, s_{i}, of each N/n_{i} (mod n_{i})
using the Extended
Euclidean Algorithm.
That is, s_{i} N/n_{i} = 1 (mod n_{i}). Also, s_{i}
N/n_{i} = 0 (mod n_{j}) for all j not equal to i.
Let e_{i} = s_{i} N/n_{i}, so e_{i}=1 (mod n_{i}),
but e_{i}=0 (mod n_{j}) for all j not equal to i.
Then x = e_{1}a_{1} + e_{2}a_{2} + e_{3}a_{3} + ... + e_{k}a_{k} is a solution to the system of equations.
Wikipedia: Chinese remainder theorem
Number Theory Glossary  in particular:
pairwise coprime  a set of integers is pairwise coprime if no two elements of the set share any factor other than 1 or 1.
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