The answer is the first and the third are true.
For the first one we know that the size of r is equal to φ(n).
That's the way the totient is defined.
It's the number of positive integers less than n that are relatively prime to n.
That's exactly how we define r.
The second one is not true.
In order for this not to be true, it would mean that there is some element where a xi mod n
is equal to axj mod n, but that's not the case, because a and n are relatively prime.
We know that these values must all be different.
The only way these could be equal is if xi is equal to xj.
That means that the sets are the same size, and this set contains numbers only up to n - 1.
This set also contains numbers from 1 to n - 1.
That's why we know that s actually contains the same elements as r.
It must be a permutation of r.
Now we can use a similar idea to what we used in the proof for Fermat's little theorem.
We can take the product of these two sets.
Since we know the contain the same elements, we know their products are also equal.
Here is what we get: the Π(R) is equal to the xi's all multiplied together,
which is equal to the Π(S), which is equal to these values all multiplied together.
We also know from this property that the number of terms is φ(n).
That's true on both sides.
That means we can separate out the a's from the x's.
We're going to have a^φ(n) times all the x's--still equal to this product.
Now we can do the division.
Removing the x's from both sides, we end up with exactly what we need,
which is that 1 is congruent to a^φ(n) mod n for any a and n where a and n are relatively prime.
