Modular Multiplicative Inverse

*Hi I do not understand the idea of Modular Multiplicative Inverse at all. I have tried reading up on modular arithmetic for a day, but I still do not understand it.*

I had the following question, which is a past olympiad question:

**N is a 4 digit number wich doesnt end in a 0, and R(N) is a 4 digit integer obtained by reversing the digits of N. So R(1997) = 7991**

Determine all such integers N for which R(N) = 4N + 3.

The answer is 1997 only. Below is the solution:

N = 1000a + 100b + 10c + d

Since R(N) is odd, a is odd. Also, 4N + 3 is 4 digits so a<3.

R(N) = 4N + 3

so:

1000d + 100c + 10b + 1 = 4000 + 400b + 40c + 4d + 3

**"now working modulo 5, we see 4d=3(mod 5) so d=2(mod 5)." **

I dont understand where this comes from... The equals signs here actually should be 3 lines, so ignore that.

So d = 2 or d = 7 etc..