Question:
How many elements of the group \(\mathbb{Z}_{50}\) have order 10?
How many elements of the group \(\mathbb{Z}_{50}\) have order 10?
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In a cyclic group \(\mathbb{Z}_n\), for any divisor \(d\) of \(n\), there are exactly \(\varphi(d)\) elements of order \(d\).
Updated On: Dec 6, 2025
- 10
- 4
- 5
- 8
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The Correct Option is B
Solution and Explanation
Step 1: Formula for order in cyclic group.
In a cyclic group \(\mathbb{Z}_n\), the number of elements of order \(d\) is given by \(\varphi(d)\), where \(\varphi\) is Euler’s totient function, provided \(d \mid n\).
Step 2: Apply the formula.
Here \(n=50\), and we want elements of order \(10\). Since \(10 \mid 50\), \[ \text{Number of elements} = \varphi(10) = \varphi(2 \times 5) = 10\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{5}\right) = 4. \]
Step 3: Conclusion.
Hence, there are 4 elements of order 10 in \(\mathbb{Z}_{50}\).
In a cyclic group \(\mathbb{Z}_n\), the number of elements of order \(d\) is given by \(\varphi(d)\), where \(\varphi\) is Euler’s totient function, provided \(d \mid n\).
Step 2: Apply the formula.
Here \(n=50\), and we want elements of order \(10\). Since \(10 \mid 50\), \[ \text{Number of elements} = \varphi(10) = \varphi(2 \times 5) = 10\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{5}\right) = 4. \]
Step 3: Conclusion.
Hence, there are 4 elements of order 10 in \(\mathbb{Z}_{50}\).
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