Write the addition and multiplication tables modulo 8 and...

write the addition and multiplication tables modulo 8, and solve the following equations in Z8: a) [4]+x=[2] b) [4]x=[0] c) [4]x=[1] d) [5]x=[1] In Zn, [1] is always the multiplicative identity. If [x][y]=[1], then [x] and [y] are multiplicative inverses. e) Which elements of Z8 have multiplicative inverses?

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00:01 We are asked to write the addition modulo 8 table.

00:09 That is z8.

00:10 And for that, we need a table of 8 rows and 8 columns.

00:15 So that will look like this.

00:19 So we shall fill this.

00:20 So this is addition modulo 8.

00:22 So it will begin with 0, 1, 2, 3, 4, 5, 6 and 7 here also.

00:31 So 0 plus 0 will give us 0.

00:34 So the first row will be as is.

00:37 And the second row will be having 5, 6, 7.

00:42 And 1 plus 7 will be 8.

00:44 But we have to write the mod.

00:46 So 8 by 8 will give us the remainder 0.

00:48 So it will be 0.

00:50 So this here, 2 plus 0 will be 2.

00:52 2 plus 1 will be 3.

00:54 So we will continue this till 7.

00:57 And 2 plus 6 is 8.

00:58 So that will give us 0.

00:59 And 2 plus 7 is 9.

01:01 9 minus 8 is 1.

01:03 So it will give us 1.

01:04 Then this will be 3 4 5 6 7 then 0 1 and then 2 similarly here it will start with 5 and 0 1 2 3 & 4 then here it will be 6 7 then 0 to 5 here it will be 7 and then 0 to 6 so this is the addition model and now we will write the multiplication model so for that we will take the similar table so we will fill it with the multiplicities so it will be 0 1 2 3 4 5 6 and 7 and similarly we will have 0 here...

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