I am working on my own version of a proof of RSA and have come do a conclusion based on these simplified statements.
Given: N = pq
X ≡ 1 (mod p)
X ≡ 1 (mod q)
X ≡ 1 (mod N) by chinese remainder theorem
I have the original expression: MX ≡ M' (mod N)
When substituting X ≡ 1 (mod N) into this equation, is it correct to say that M = M' (removing the modulus) or must I say M ≡ M' (mod N)?