 So here's another example of forming a group of integers mod n. So while we can use either integer addition or multiplication to form the group of integers mod n, and again in a little while we'll actually use both of these, one problem is that if we use multiplication of the element cn we might not be able to find inverses for all elements and that means we won't be able to use all of the elements in a group. So what that means is that if we want to talk about the group of integers mod n, it's convenient to start with the operation of addition only and then we can talk about the group zn. And this group will consist of the integers mod n where the operation that we'll be using is the operation of integer addition. For example, suppose we want to find the Cayley table for the group z6 and so we begin by looking at the distinct values mod 6. These are going to be the possible remainders when a number is divided by 6, namely 0, 1, 2, 3, 4, or 5. We form the Cayley table. 0 is the additive identity and so this allows us to fill in that row and column. And then we can do the other addition remembering that a is congruent, a is equivalent to b, whenever a and b have the same remainders when divided by 6. So 1 plus 1 is 2. 1 plus 2 is 3. 1 plus 3 is 4. 1 plus 4 is 5. 1 plus 5 is 6. But there is no element 6. That means we're going to have to replace 6 with something. To find that something we need to find a number with the same remainder when divided by 6. Well, when 6 is divided by 6, the remainder is 0, so that means that 6 and 0 are going to be congruent. And so we have 1 plus 5 is congruent to 0. Incidentally, this also means that 5 is the inverse of 1. Now because the operation is the operation of addition, we also know that addition is commutative, so that set of sums also gives us another set of sums. So we can add 2. So 2 plus 2 is 4. 2 plus 3 is 5. 2 plus 4 is 6, which is congruent to 0 again. 2 plus 5 is 7. And again, there is no 7, but 7 and 1 are congruent because they both have remainder 1 when divided by 6. And again, addition is commutative, so these sums give us another set of sums. 3 plus 2 plus 2, 3 plus 2, 4 plus 2, and 5 plus 2. And we continue with the addition. 3 plus 3 is 6, which is congruent to 0. 3 plus 4 is 7, which is congruent to 1. 3 plus 5 is 8, which is congruent to 2. 4 plus 4 is 8, which is congruent to 2. 4 plus 5 is 9, congruent to 3. 5 plus 5 is 10, which is congruent to 4. And that completes our Cayley table.