$e = 503456131$ is a prime number. It is relatively prime to the number $b = 10000123400257488$
If I use the extended euclidean algorithm (using this python implementation) to calculate the coefficients on the linear combination of e and b that gives 1, I obtain $-906226286492069\cdot e + 45623955 \cdot b = 1$.
Is it not correct that the coefficient on e, namely $-906226286492069$, is a multiplicative inverse of e (mod b)?
I thought this was correct, but $-906226286492069\cdot e$ divided by b gives remainder $10000096$.
What am I not seeing here?