chineseremaindertheorem
Chinese Remainder Theorem
Let hcf (n1,n2,n3,…,nr)=1. Then the system of linear congruences:
x==a1 [mod n1]
x==a2 [mod n2]
x==a3 [mod n3] …
x==ar [mod nr]
has a simultaneous solution, unique modulo n1n2n3…nr
Proof:
Let n=n1n2n3…nr
Let Nk = n/nk, and Nkxk == 1 [mod nk]
Let X=a1N1x1 + a2N2x2 + … + arNrxr
Then, Ni==0 [mod nk][i<>k]
so X == akNkxk == ak.1 == ak [mod nk]
Suppose Y is another solution:
X==Y [mod nk]
=> nk|(X-Y) for each k
But, hcf (ni,nj)=1 => n1n2n3…nr|(X-Y)
=> X==Y [mod n]