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How many elements of the group $\mathbb{Z}_{50}$ have order 10?

(A) 10

(B) 4

(C) 5

(D) 8
in MA2021 by Professor | 63 views

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In a cyclic group of order $n$, the number of elements that have order $k$ is given by $\phi(k)$, where $\phi(k)$ is the Euler's totient function evaluated at $k$.

So, the correct number of elements in a cyclic group of order $n$ that have order $k$ is $\phi(k)$.

In the context of the specific question, we are dealing with the cyclic group $\mathbb{Z}_{50}$ of order 50. We want to find elements of order 10, so we need to calculate $\phi(10)$.

Using Euler's totient function, $\phi(10) = 4$.

Therefore, the correct number of elements of order 10 in $\mathbb{Z}_{50}$ is 4.

The correct answer is:

(B) 4.
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