Assume that [a],[b] is an element of (Z12 ,circle plus), order of ([a])=m, order of ([b])=n. Find examples of [a] and [b] (one example for each case, so three examples) for which order of ([a] circle plus [b])=mn, order of ([a] circle plus [b])=lcm(m,n) doesnt equal mn, and order of ([a] circle plus [b]) doesnt equal mn and order of ([a] circle plus [b]) doesnt equal lcm(m,n) Assume that $[a],[b] \in \mathbb{Z}_{12}^{\times}$, $o([a]) = m$, $o([b]) = n$.Find examples of $[a]$ and $[b]$ (one example for each case, so three examples) for which:1. $o([a] \cdot [b]) = mn$,2. $o([a] \cdot [b]) = \text{lcm}(m,n) \neq mn$,3. $o([a] \cdot [b]) \neq mn$ and $o([a] \cdot [b]) \neq \text{lcm}(m,n)$.(Note: $o([a])$ denotes the order of $[a]$ in the group $\mathbb{Z}_{12}^{\times}$, and $\text{lcm}(m,n)$ denotes the least common multiple of $m$ and $n$.)

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Answered: Assume that [a],[b]=( Z12,,5), o([a]) =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Assume that [a],[b] is an element of (Z₁₂ ,circle plus), order of ([a])=m, order of ([b])=n. Find examples of [a] and [b] (one example for each case, so three examples) for which order of ([a] circle plus [b])=mn, order of ([a] circle plus [b])=lcm(m,n) doesnt equal mn, and order of ([a] circle plus [b]) doesnt equal mn and order of ([a] circle plus [b]) doesnt equal lcm(m,n)

Assume that $[a],[b] \in \mathbb{Z}_{12}^{\times}$, $o([a]) = m$, $o([b]) = n$.

Find examples of $[a]$ and $[b]$ (one example for each case, so three examples) for which:

1. $o([a] \cdot [b]) = mn$,
2. $o([a] \cdot [b]) = \text{lcm}(m,n) \neq mn$,
3. $o([a] \cdot [b]) \neq mn$ and $o([a] \cdot [b]) \neq \text{lcm}(m,n)$.

(Note: $o([a])$ denotes the order of $[a]$ in the group $\mathbb{Z}_{12}^{\times}$, and $\text{lcm}(m,n)$ denotes the least common multiple of $m$ and $n$.)

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