To find the order of $a^{14}$ where the order of $a$ is 60, we can use the following steps:

Step 1: Recall the Definition of the Order of an Element

The order of an element $g$ in a group is the smallest positive integer $n$ such that $g^n = e$, where $e$ is the identity element.

Step 2: Calculate the Order of $a^{14}$

If the order of $a$ is 60, this means: $a^{60} = e$ Now, to find the order of $a^{14}$, we need to find the smallest positive integer $k$ such that: $(a^{14})^k = e$ This simplifies to: $a^{14k} = e$ Since the order of $a$ is 60, we have $a^{60} = e$, so $14k$ must be a multiple of 60. Therefore, $k$ is the smallest positive integer such that: $14k \equiv 0 \pmod{60}$ This implies that $k$ is the smallest positive integer such that $14k$ is divisible by 60.

Step 3: Solve for $k$

To find $k$, we calculate the least common multiple (LCM) of 14 and 60:

1. The prime factorization of 14 is $14 = 2 \times 7$.
2. The prime factorization of 60 is $60 = 2^2 \times 3 \times 5$.

The LCM of 14 and 60 is: $\text{LCM}(14, 60) = 2^2 \times 3 \times 5 \times 7 = 420$ Thus, $14k = 420$, which gives: $k = \frac{420}{14} = 30$ Therefore, $k = 30$.

Answer:

The order of $a^{14}$ is 30.

Do you want details or have any questions?

Here are some related questions:

1. What is the order of $a^k$ in general when the order of $a$ is $n$?
2. Can you explain why the order of $a^{14}$ divides 60?
3. How do you find the LCM of two numbers using prime factorization?
4. What if the order of $a$ was a prime number? How would that affect $a^{14}$?
5. How would you verify that the order of $a^{14}$ is indeed 30?
6. Can you find the order of $a^7$ given that the order of $a$ is 60?
7. How does the concept of order of elements relate to cyclic subgroups?
8. What is the significance of orders in the context of group theory?

Tip: The order of an element $a^k$ in a group is given by the order of $a$ divided by the greatest common divisor (gcd) of $k$ and the order of $a$.