$G=(\{0,1,2,3,4,5\},+_{6})$ is a group. Order of 5 is:

Q: $G=(\{0,1,2,3,4,5\},+_{6})$ is a group. Order of 5 is:

To determine the order of the element 5 in the group $ G = (\{0, 1, 2, 3, 4, 5\}, +_6) $, we need to find the smallest positive integer $ n $ such that $ n \cdot 5 \equiv 0 \pmod{6} $. This means we are looking for the smallest $ n $ such that when 5 is added to itself $ n $ times, the result is congruent to 0 modulo 6. Let's calculate: 1. $ 1 \cdot 5 = 5 \equiv 5 \pmod{6} $ 2. $ 2 \cdot 5 = 10 \equiv 4 \pmod{6} $ 3. $ 3 \cdot 5 = 15 \equiv 3 \pmod{6} $ 4. $ 4 \cdot 5 = 20 \equiv 2 \pmod{6} $ 5. $ 5 \cdot 5 = 25 \equiv 1 \pmod{6} $ 6. $ 6 \cdot 5 = 30 \equiv 0 \pmod{6} $ From these calculations, we see that the smallest $ n $ for which $ n \cdot 5 \equiv 0 \pmod{6} $ is $ n = 6 $. Therefore, the order of the element 5 in this group is 6. Now, let's explain why the other options are incorrect: - **Option 1 (Order 1):** If the order were 1, then $ 1 \cdot 5 \equiv 0 \pmod{6} $. However, $ 1 \cdot 5 = 5 \equiv 5 \pmod{6} $, not 0. Therefore, the order cannot be 1. - **Option 2 (Order 3):** If the order were 3, then $ 3 \cdot 5 \equiv 0 \pmod{6} $. However, $ 3 \cdot 5 = 15 \equiv 3 \pmod{6} $, not 0. Therefore, the order cannot be 3. - **Option 3 (Order 5):** If the order were 5, then $ 5 \cdot 5 \equiv 0 \pmod{6} $. However, $ 5 \cdot 5 = 25 \equiv 1 \pmod{6} $, not 0. Therefore, the order cannot be 5. Thus, the only correct answer is Option 4, which states that the order of 5 is 6.

Question

Accepted Answer

To determine the order of the element 5 in the group $ G = (\{0, 1, 2, 3, 4, 5\}, +_6) $, we need to find the smallest positive integer $ n $ such that $ n \cdot 5 \equiv 0 \pmod{6} $. This means we are looking for the smallest $ n $ such that when 5 is added to itself $ n $ times, the result is congruent to 0 modulo 6.

Let's calculate:

1. $ 1 \cdot 5 = 5 \equiv 5 \pmod{6} $
2. $ 2 \cdot 5 = 10 \equiv 4 \pmod{6} $
3. $ 3 \cdot 5 = 15 \equiv 3 \pmod{6} $
4. $ 4 \cdot 5 = 20 \equiv 2 \pmod{6} $
5. $ 5 \cdot 5 = 25 \equiv 1 \pmod{6} $
6. $ 6 \cdot 5 = 30 \equiv 0 \pmod{6} $

From these calculations, we see that the smallest $ n $ for which $ n \cdot 5 \equiv 0 \pmod{6} $ is $ n = 6 $. Therefore, the order of the element 5 in this group is 6.

Now, let's explain why the other options are incorrect:

- **Option 1 (Order 1):** If the order were 1, then $ 1 \cdot 5 \equiv 0 \pmod{6} $. However, $ 1 \cdot 5 = 5 \equiv 5 \pmod{6} $, not 0. Therefore, the order cannot be 1.

- **Option 2 (Order 3):** If the order were 3, then $ 3 \cdot 5 \equiv 0 \pmod{6} $. However, $ 3 \cdot 5 = 15 \equiv 3 \pmod{6} $, not 0. Therefore, the order cannot be 3.

- **Option 3 (Order 5):** If the order were 5, then $ 5 \cdot 5 \equiv 0 \pmod{6} $. However, $ 5 \cdot 5 = 25 \equiv 1 \pmod{6} $, not 0. Therefore, the order cannot be 5.

Thus, the only correct answer is Option 4, which states that the order of 5 is 6.