I am trying to understand how this method for simplifying the decryption process works. I understand that if I have the message c raised to the public exponent modulo the modulus, then to decrypt it, I need to raise the encrypted message to the private exponent modulo the modulus.

Say I have a public key (13, 673627) and a private key (103381, 673627) and I want to read the message 174277, I understand that I can do this by raising $\displaystyle 174277^{103381}$ and taking mod 673627 to get it. However, I have a guided question that goes through a method that doesn't require as much computation and I don't quite understand it.

It says "find the other information in (103381, 673627), $\displaystyle e_1$ is chosen so that $\displaystyle 13e_1 \equiv 1~mod~ p_1$, $\displaystyle e_2$ chosen so that $\displaystyle 13e_2 \equiv 1~ mod~ p_2$ and $\displaystyle c$ chosen so that $\displaystyle p_1c\equiv 1~ mod~ p_2$.

I factored the modulus to get 673627 = 919*733 = $\displaystyle p_1p_2$

I found that:

$\displaystyle 13(707) \equiv 1~ mod~ 919 $

and

$\displaystyle 13(282)\equiv 1~ mod~ 733$

and

$\displaystyle 919(67)\equiv 1~ mod~ 733$

How do I use them to decrypt 174277?