Order of an Element

The order of an element in a group is the smallest positive integer 'n' such that the element raised to the power of 'n' gives the identity element of the group. In other words, it is the smallest positive integer 'n' for which a^n ≡ 1 (mod m), where 'a' is the element and 'm' is the modulus.

Multiplicative Group of Integers Modulo 5

To find the order of an element in the multiplicative group of integers modulo 5, we need to consider the set of numbers {1, 2, 3, 4} under multiplication modulo 5.

Finding the Order of {3}

To find the order of {3}, we need to calculate powers of {3} modulo 5 until we find {1}, which is the identity element.

Powers of {3} modulo 5:
- {3}^1 ≡ 3 (mod 5)
- {3}^2 ≡ 4 (mod 5)
- {3}^3 ≡ 2 (mod 5)
- {3}^4 ≡ 1 (mod 5)

Explanation:
- We start by calculating {3} raised to the power of 1 modulo 5, which gives us 3.
- Next, we calculate {3}^2 modulo 5, which is 4.
- Continuing, we calculate {3}^3 modulo 5, which is 2.
- Finally, we calculate {3}^4 modulo 5, which is 1. This means that {3}^4 is congruent to 1 modulo 5.

Order of {3}
The order of {3} in the multiplicative group of integers modulo 5 is the smallest positive integer 'n' such that {3}^n ≡ 1 (mod 5). In this case, 'n' is 4, as we found that {3}^4 ≡ 1 (mod 5). Therefore, the correct answer is '4'.