G is a group with a being an element of G. The order of a is 60, what is the order of a^14?

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 \( a \).

Explore how to calculate the order of a^14 where the order of a in a group G is 60. Understand the concept of order in group theory, apply modular arithmetic, and use Lagrange's theorem. Detailed step-by-step solution provided.

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