Find all residues (a) such that (a) is its own inverse modulo 317.

simpler version: find all numbers (a) that equal its own inverse modulo 317

To solve for the value of the residues A, we can use the formula:

A^2 = 1                 (mod prime)

This is only true for prime numbers. The given number which is 317 is a prime number therefore the values of the residues A are:

A = + 1, - 1

Since I believe the problem specifically states for the list of positive integers only and less than 317, a value of A = - 1 is therefore not valid. However, a value of – 1 in this case would simply be equal to:

317 – 1 = 316

Therefore the residue values A are 1 and 316.