Example

Given $a \;\equiv\; 2$ (mod 5)
$a \;\equiv\; 3$ (mod 11)
Find $a \;\equiv\; x$ (mod 55)

a₁ = 2, a₂ = 3
m₁ = 11, m₂ = 5
n₁ = 5, n₂ = 11
m₁^-1 mod n₁ = 11^-1 mod 5 = 1
m₂^-1 mod n₂ = 5^-1 mod 11 = 9
$c_1 \;=\; m_1(m_1^{-1} \;{\rm mod}\; n_1)$ = 11(1) = 11
$c_2 \;=\; m_2(m_2^{-1} \;{\rm mod}\; n_2)$ = 5(9) = 45

a $\equiv$ 2*11 + 3*45 (mod 55)
$\equiv$ 22 + 135 (mod 55)
$\equiv$ 157 (mod 55)
$\equiv$ 47 (mod 55)