# Problem 451: Modular inverses

![graphic](img451.gif)

Consider the number 15. There are eight positive numbers less than 15
which are coprime to 15: 1, 2, 4, 7, 8, 11, 13, 14. The modular inverses
of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14 because 1\*1
mod 15=1 2\*8=16 mod 15=1 4\*4=16 mod 15=1 7\*13=91 mod 15=1 11\*11=121
mod 15=1 14\*14=196 mod 15=1 Let I(n) be the largest positive number m
smaller than n-1 such that the modular inverse of m modulo n equals m
itself. So I(15)=11. Also I(100)=51 and I(7)=1. Find ∑I(n) for 3≤n≤2·107