Compute the inverse of an integer modulo 100000000003
Pyth, 24 bytes

L-b*/bJ+3^T11Jy*uy^GT11Q


Test suite

This uses the fact that a^(p-2) mod p = a^-1 mod p.

First, I manually reimplement modulus, for the specific case of mod 100000000003. I use the formula a mod b = a - (a/b)*b, where / is floored division. I generate the modulus with 10^11 + 3, using the code +3^T11, then save it in J, then use this and the above formula to calculate b mod 100000000003 with -b*/bJ+3^T11J. This function is defined as y with L.

Next, I start with the input, then take it to the tenth power and reduce mod 100000000003, and repeat this 11 times. y^GT is the code executed in each step, and uy^GT11Q runs it 11 times starting with the input.

Now I have Q^(10^11) mod 10^11 + 3, and I want Q^(10^11 + 1) mod 10^11 + 3, so I multiply by the input with *, reduce it mod 100000000003 with y one last time, and output.