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## Find the generators of Z/(48)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.3

Find all the generators in $\mathbb{Z}/(48)$.

Solution: The generators of $\mathbb{Z}/(48)$ are precisely those (equivalence classes represented by) integers $k$, $1 \leq k \leq 48$, such that $\mathsf{gcd}(k,48) = 1$. Since $48$ factors as $48 = 2^4 \cdot 3$, we eliminate precisely those integers which are multiples of 2 or 3. This leaves as generators the integers 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, and 47.