## Retractions C

### Problem 447

Published on Saturday, 16th November 2013, 10:00 pm; Solved by 120; Difficulty rating: 95%
For every integer n>1, the family of functions f_{n,a,b} is defined
by f_{n,a,b}(`x`)≡a`x`+b mod n for a,b,`x` integer and 0<a<n, 0≤b<n, 0≤`x`<n.

We will call f_{n,a,b} a *retraction* if f_{n,a,b}(f_{n,a,b}(`x`))≡f_{n,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(10^{7})≡638042271 (mod 1 000 000 007).

Find F(10^{14}) (mod 1 000 000 007).