Find all residues $a$ such that $a$ is its own inverse modulo $317.$ (Your answer should be a list of integers greater than 0 and less than $317,$ separated by commas.)
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.