Fermat's Little Theorem states that a^{n-1}=/= 1 mod n if a =/= n

I've uncovered some empirical evidence to suggest that if a =/= b =/= n then a^{n-2}=/= b^{n-2}mod n.

Consider the following "mod 5 matrix".

2_{5}3_{5}4_{5}5_{5}

1_{5}1 1 1 1 For clarification, 2_{5}represents all the numbers that are congruent to 2 mod 5

2_{5}4 3 1 2 4_{5 }represents all the numbers that are congruent to 4 mod 5

3_{5}4 2 1 3 For example 2^{3}== 3 mod 5 or 7^{8 }== 3 mod 5

4_{5}1 4 1 4 Note that the next to the next to the last column confirms Fermat's Little Theorem, and the last column

5_{5}0 0 0 0 confirms that a == a^{n}mod n

If we construct a similar matrix for mod 7, mod 11 and mod 13, we will observe the same correlations. In addition we will observe that

all of the congruence values in the third from the last column (equivalent to a^{n-2}) are unique with no two the same. If you construct a matrix for

an even number or a non-prime number the congruence values are not unique. These observations lead me to believe that:

If a =/= b =/=n then a^{n-2}=/= b^{n-2}mod n.

I would appreciate any advice or assistance in proving this conjecture.