Beautiful Numbers - Codeforces problem doubt

The problem is this:

http://codeforces.com/problemset/problem/300/C

Thanks to it, I managed to understand a bit better the usage and concept of modular multiplicative inverse.

Namely, if we want to compute x^(-1) % (M=a big prime number) then we can use Euler's Theorem and do:

x^(M-2) % M and compute it by fast exponentiantion, is this so?

However, many contestants there used this function:

long long int big_mod(long long int i, long long int j)
{
    if(j==0)
        return 1;
    long long int x=big_mod(i,j>>1);
    x=(x*x)%mod;
    if(j&1)
        return (x*i)%mod;
    return x;
}

My doubt is:

Is this function doing the same job as the standard fast exponentiation function? If not, why?

Thanks in advance,

Bruno