Question

Let ℤ17 be the ring of integers modulo 17, and let ℤ×17 be the group of units of ℤ17 under multiplication. Which of the following are generators of ℤ×17? I.5 II.8 III.16

(I) 5.
(II) 8.
(III) 16.

(A) None.
(B) I only.
(C) II only.
(D) III only.
(E) I, II, and III

Answer 1

Answer: The correct option is (E) I, II and III.

Step-by-step explanation: Given that ℤ17 be the ring of integers modulo 17, and let ℤ×17 be the group of units of ℤ17 under multiplication.

We are to select the correct generators of ℤ×17.

According to the given information, we can write

ℤ17 = {0, 1, 2, 3, . . . , 16}

and

ℤ×17 = {1, 2 , 3 , . . . 16}.

The generators of ℤ×17 are those elements which are co prime to 17.

Since all the elements from 1 to 16 are co prime to 17, so all the elements of the set ℤ×17 are its generators.

Therefore, 5, 8 and 16, all are generators. So, i, ii and iii are generators.

Thus, (E) is the correct option.