Order of an Element in a Group
In group theory, the order of an element refers to the smallest positive integer 'n' such that raising the element to the power of 'n' gives the identity element of the group. In other words, it is the smallest exponent that returns the element to its original state.

Group U(15)
The group U(15) consists of all positive integers less than 15 that are coprime to 15. In other words, U(15) contains all the positive integers between 1 and 15 that do not have any common factors with 15, except for 1.

Calculating the Order of 13 in U(15)
To find the order of the element 13 in the group U(15), we need to calculate the smallest positive integer 'n' such that 13^n ≡ 1 (mod 15), where '≡' denotes congruence modulo 15.

We can calculate the powers of 13 modulo 15 and observe the pattern:

13^1 ≡ 13 (mod 15)
13^2 ≡ 4 (mod 15)
13^3 ≡ 7 (mod 15)
13^4 ≡ 1 (mod 15)

As we can see, 13^4 ≡ 1 (mod 15), which means the order of 13 in U(15) is 4. This implies that raising 13 to any multiple of 4 will result in 1 modulo 15.

Explanation
By calculating the powers of 13 modulo 15, we observe that the sequence eventually repeats itself. In this case, it repeats after 4 powers. This is because 13 and 15 are coprime, and the order of any element in U(15) is always a divisor of the order of the group itself, which is φ(15) = 8, where φ denotes Euler's totient function.

Since the order of 13 is 4, it means that 13^4 ≡ 1 (mod 15), and any higher power of 13 can be reduced to a power equivalent to 1 modulo 15. Therefore, the order of 13 in the group U(15) is 4.

Conclusion
The order of the element 13 in the group U(15) is 4. This means that raising 13 to the power of 4 gives the identity element of the group. The order can be determined by calculating the powers of 13 modulo 15 and observing the repeating pattern.