Retractions B

Published on Saturday, 16th November 2013, 10:00 pm; Solved by 261;
Difficulty rating: 60%

Problem 446

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(n4+4) for 1≤n≤N.

Find F(107) (mod 1 000 000 007)