Exercise · ZW-ORDER-OF-A-GROUP-009 · ★★☆ Intermediate
What is the order of the group of units mod 100?
Choose one option.
Show answer
- A. 40
Explanation
Since \(100=2^2\cdot5^2\), \(\varphi(100)=100\cdot\tfrac12\cdot\tfrac45=40\). Counting only odd numbers would give 50, but multiples of 5 must also be excluded.
Review the concept: Order of a group
The order of a finite group \(G\) is its number of elements; if \(G\) is infinite it has infinite order.
\[|G|\]
More practice problems: order of a group
- ZW-ORDER-OF-A-GROUP-001 What is the order of the group (ℤ_9,+)?
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- ZW-ORDER-OF-A-GROUP-003 What is the order of the group (ℤ,+)?
- ZW-ORDER-OF-A-GROUP-004 The group of units mod 12, \x : gcd(x,12)=1\ under multiplication mod 12, has order
- ZW-ORDER-OF-A-GROUP-005 What is the order of the group of units mod 15?
- ZW-ORDER-OF-A-GROUP-006 The rotations that leave a square in the same position form a group. What is its order?
- ZW-ORDER-OF-A-GROUP-007 What is the smallest order a group can have?
- ZW-ORDER-OF-A-GROUP-008 How many elements does the group GL_2(ℤ_2) of invertible 2×2 matrices with entries in ℤ_2 have?