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.