Yes, you're correct! The greatest common divisor (gcd) of 26 and 12 is indeed 2.

We can calculate this using the Euclidean algorithm:

1. $26 \div 12 = 2$ remainder $2$.
2. $12 \div 2 = 6$ remainder $0$.

When the remainder reaches 0, the last non-zero remainder is the gcd. Therefore: $\gcd(26, 12) = 2.$

This gcd value was used to determine the order of the element $26 + \langle 12 \rangle$ in the factor group $\mathbb{Z}_{60} / \langle 12 \rangle$.

Would you like more details or examples on how the gcd impacts the order of elements in group theory?

Here are 5 related questions:

1. What is the Euclidean algorithm and how does it work?
2. Why does the gcd play a role in finding the order of an element in modular arithmetic?
3. How do we find the lcm (least common multiple) using the gcd?
4. What are some applications of the gcd in number theory?
5. How does the gcd influence the structure of cyclic groups?

Tip: The gcd not only helps with solving modular problems but is also essential in simplifying fractions and solving Diophantine equations!