Answer: The calculations are done below.
Step-by-step explanation: We are given to find the values for each of the following :
(A) 3 times 5 in [tex]\mathbb{Z}_{10}.[/tex]
We know that
[tex]\mathbb{Z}_{10}=\{0,1,2,3,4,5,6,7,8,9\}(\textup{mod}10).[/tex]
Therefore, we get
[tex]3\times5=15(\textup{mod}10)=5.[/tex]
(B) 3 times 5 in [tex]\mathbb{Z}_{5}.[/tex]
We know that
[tex]\mathbb{Z}_{5}=\{0,1,2,3,4\}(\textup{mod}5).[/tex]
Therefore, we get
[tex]3\times5=15(\textup{mod}5)=0.[/tex]
(C) 3 times 5 in [tex]\mathbb{Z}_{3}.[/tex]
We know that
[tex]\mathbb{Z}_{3}=\{0,1,2\}(\textup{mod}3).[/tex]
Therefore, we get
[tex]3\times5=15(\textup{mod}3)=0.[/tex]
(D) 5 times 3 in [tex]\mathbb{Z}_{10}.[/tex]
We know that
[tex]\mathbb{Z}_{10}=\{0,1,2,3,4,5,6,7,8,9\}(\textup{mod}10).[/tex]
Therefore, we get
[tex]5\times3=15(\textup{mod}10)=5.[/tex]
Thus, the required values are evaluated.
