Use the Chinese remainder theorem to solve the system of lin

Use the Chinese remainder theorem to solve the system of linear congruences x 3 mod 5 x 6 mod 7 x 4 mod 11.

Solution

Since GCF(5,7)=GCF(5,11)=GCF(7,11) = 1

N = n₁.n_2.n₃ = 5*11*7 = 385

Also C_i = N/n_i

=> C₁ = 385/5 = 77, Similarly C_{2 =} 55 and C_{3 =} 35

=> 5X₁+77Y₁ = 1 ( Y₁ = -2), 7X₂+55Y₂ = 1 (Y_{2 =} -1) and 11X₃+35Y₃ = 1 (Y₃ = -5)

Since We have

X = a₁₍₇₇Y₁₎₊a₂₍₅₅Y₂₎+a₃₍₃₅Y_{3) = -462-330-700 = -1492}