Binomial coefficient modulo 142857

Question

How to calculate binomial coefficient modulo 142857 for large n and r. Is there anything special about the 142857? If the question is modulo

Answer 1

What's special about 142857 is that 7 * 142857 = 999999 = 10^6 - 1. This is a factor that arises from Fermat's Little Theorem with a=10 and p=7, yielding the modular equivalence 10^7 == 10 (mod 7). That means you can work modulo 999999 for the most part and reduce to the final modulus by dividing by 7 at the end. The advantage of this is that modular division is very efficient in representation bases of the form 10^k for k=1,2,3,6. All you do in such cases is add together digit groups; this is a generalization of casting out nines.

This optimization only really makes sense if you have hardware base-10 multiplication. Which is really to say that it works well if you have to do this with paper and pencil. Since this problem recently appeared on an online contest, I imagine that's exactly the origin of the question.