Problem 445: Retractions A

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For every integer n>1, the family of functions fn,a,b is defined

by fn,a,b(x)≡ax+b mod n for a,b,x integer and 0

You are given that ∑ R(c) for c=C(100 000,k), and 1 ≤ k ≤99 999 ≡628701600 (mod 1 000 000 007). (C(n,k) is the binomial coefficient).

Find ∑ R(c) for c=C(10 000 000,k), and 1 ≤k≤ 9 999 999. Give your answer modulo 1 000 000 007.