RSA暗号
よって
a ′ ≡ ( a λ ( n ) ) x ⋅ a ( mod p ) ≡ ( a λ ( n ) ) x ⋅ 0 ( mod p ) = 0 a ′ ≡ ( a λ ( n ) ) x ⋅ a ( mod q ) ≡ 1 x ⋅ a ( mod q ) = a {\displaystyle {\begin{aligned}a'\equiv (a^{\lambda (n)})^{x}\cdot a{\pmod {p}}\equiv (a^{\lambda (n)})^{x}\cdot 0{\pmod {p}}=0\\a'\equiv (a^{\lambda (n)})^{x}\cdot a{\pmod {q}}\equiv 1^{x}\cdot a{\pmod {q}}=a\end{aligned}}}
よって
a ′ ≡ 0 ( mod p ) ≡ a ( mod p ) a ′ ≡ a ( mod q ) {\displaystyle {\begin{aligned}a'\equiv 0{\pmod {p}}\equiv a{\pmod {p}}\\a'\equiv a{\pmod {q}}\end{aligned}}}
よって a' − a は p, q で割り切れるから
a ′ ≡ a ( mod n ) {\displaystyle a'\equiv a{\pmod {n}}}