0:00
Hello.
00:01
So first, i'm finding the order of the element 1 bar.
00:04
Well, it's clearly the order of 1 is 1, since 1 is the identity.
00:16
Now, since we have that negative 1 is going to be equal to 11 mod 12.
00:25
So we have that negative 1 is an element of the equivalence class of 11.
00:31
So 11 bar.
00:32
Now, from the definition of the equivalence relation here, so negative 1 is equal, with negative 1 bar is equal to 11 bar, so in the same equivalence class, and we know that 11 bar, so squared here is equal to 11 times 11, so that's equal to 121, but 121 is equal to 1 mod 12, right, because the remainder here is 1.
01:03
1 .21 is equal to 1.
01:06
1 .12.
01:06
So therefore, we have that the order of the element of negative 1 bar is going to be 2.
01:19
And then for n bar being equal to 5 bar, we have that the equivalence class here...