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Compute the number of generators of Z/(49000)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.5

Find the number of generators for $\mathbb{Z}/(49000)$.

Solution: The number of generators of $\mathbb{Z}/(n)$ is $\varphi(n)$, where $\varphi$ denotes the Euler totient function.

Now $49000 = 2^3 \cdot 5^3 \cdot 7^2$. Thus $$\varphi(49000) = \varphi(2^3)\varphi(5^3)\varphi(7^2) = 2^2 (2-1) 5^2 (5-1) 7 (7-1) = 16800.$$


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