Problem 447Published on Saturday, 16th November 2013, 10:00 pm; Solved by 134; Difficulty rating: 95%
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<a<n, 0≤b<n, 0≤x<n.
We will call fn,a,b a retraction if fn,a,b(fn,a,b(x))≡fn,a,b(x) mod n for every 0≤x<n.
Let R(n) be the number of retractions for n.
F(N)=∑R(n) for 2≤n≤N.
F(107)≡638042271 (mod 1 000 000 007).
Find F(1014) (mod 1 000 000 007).