Math Problem Statement

Order of the element 7 in the multiplicative group of 21

Solution

To find the order of the element 77 in the multiplicative group of integers modulo 2121 (denoted as Z21\mathbb{Z}_{21}^*), we need to determine the smallest positive integer kk such that:

7k1(mod21)7^k \equiv 1 \pmod{21}

Let's calculate 7kmod217^k \mod 21 for increasing values of kk:

  1. 71=77(mod21)7^1 = 7 \equiv 7 \pmod{21}
  2. 72=4949mod21=7(mod21)7^2 = 49 \equiv 49 \mod 21 = 7 \pmod{21}
  3. 73=7×72=343343mod21=7(mod21)7^3 = 7 \times 7^2 = 343 \equiv 343 \mod 21 = 7 \pmod{21}
  4. Continuing this way, it's clear that 7k7(mod21)7^k \equiv 7 \pmod{21} for any kk.

Since 7k7(mod21)7^k \equiv 7 \pmod{21} never equals 1 for any positive integer kk, the element 7 does not have an order in this group. This means that 7 is not an element of the multiplicative group Z21\mathbb{Z}_{21}^*.

Conclusion: In the multiplicative group Z21\mathbb{Z}_{21}^*, the element 77 is not invertible, and thus it has no order.

Would you like further details or have any questions?


Here are some related questions:

  1. What is the definition of the order of an element in a group?
  2. How do you determine if an element is invertible in a multiplicative group modulo nn?
  3. What is the structure of the group Z21\mathbb{Z}_{21}^*?
  4. Can every integer modulo nn have an order in Zn\mathbb{Z}_n^*?
  5. What are the other elements in Z21\mathbb{Z}_{21}^* and their orders?

Tip: To check if an element is invertible in Zn\mathbb{Z}_n^*, calculate the greatest common divisor (GCD) of the element and nn. 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