Let S = {0, 2, 4, 6, 8} Z10 with addition and multiplication as defined in Z10. Construct addition and multiplication tables for S, using the operations as defined in Z10.

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You need to evaluate the addition table in `Z_10 = {0,1,2,3,4,5,6,7,8,9}` , using modular arithmetic, hence, you need to add the elements of `S = {0, 2, 4, 6, 8}` and then, if the result is larger than 10, you need to divide it by 10. The result of the sum is the reminder that you get dividing by 10, such that:

`0 + 2 = 2 in Z_10`

`0 + 4 = 4 in Z_10`

`0 + 6 = 6 in Z_10`

`0 + 8 = 8 in Z_10`

`2 + 4 = 6 in Z_10`

`2 + 6 = 8 in Z_10`

`2 + 8 = 10`

Since `10 !in Z_10` , you need to divide by 10, such that:

`10:10 = 1 ` (reminder 0) => `2 + 8 = 10 = 0 in Z_10`

`4 + 6 = 10` = `0 in Z_10`

`4 + 8 = 12 : 10 = 1 ` reminder `2 in Z_10`

`6 + 8 = 14 : 10 = 1` reminder `4 in Z_10`

Hence, evaluating the addition table in `Z_10` yields `S = {0,2,4,6,8}.`

You need to evaluate the multiplication table in `Z_1` 0, such that:

`0*2 = 0 in Z_10`

`0*4 = 0 in Z_10`

`0*6 = 0 in Z_10`

`0*8 = 0 in Z_10 `

`2*4 = 8 in Z_10`

`2*6 = 12 = 2 in Z_10`

`2*8 = 16 = 6 in Z_10`

`4*6 = 24:10 = 2` reminder `4 in Z_10`

`4*8 = 32:10 = 3` reminder `2 in Z_10`

`6*8 = 48:10 = 4` reminder `8 in Z_10`

Hence, evaluating the multiplication table in `Z_10` yields `S = {0,2,4,6,8}.`

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