Math Problem Statement
Order of the element 7 in the multiplicative group of 21
Solution
To find the order of the element in the multiplicative group of integers modulo (denoted as ), we need to determine the smallest positive integer such that:
Let's calculate for increasing values of :
- Continuing this way, it's clear that for any .
Since never equals 1 for any positive integer , the element 7 does not have an order in this group. This means that 7 is not an element of the multiplicative group .
Conclusion: In the multiplicative group , the element is not invertible, and thus it has no order.
Would you like further details or have any questions?
Here are some related questions:
- What is the definition of the order of an element in a group?
- How do you determine if an element is invertible in a multiplicative group modulo ?
- What is the structure of the group ?
- Can every integer modulo have an order in ?
- What are the other elements in and their orders?
Tip: To check if an element is invertible in , calculate the greatest common divisor (GCD) of the element and . If the GCD is 1, the element is invertible.
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Math Problem Analysis
Mathematical Concepts
Group Theory
Modular Arithmetic
Formulas
Modular exponentiation
Theorems
Order of an element in a group
Suitable Grade Level
Advanced High School
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