Questions: Units in Zn. Let n be a positive integer. Let [a] in Zn. (a) Let [a] in Zn with gcd(a, n)=1. Using Bezout's Identity we can find integers x and y so that ax+ny=1. Explain how [a] and [x] are related in Zn. (b) Prove that if gcd(a, n)=1, then [a] is a unit in Zn. (c) Prove that if [a] is a unit in Zn, then gcd(a, n)=1. (Hint: Is the converse of Bezout's Identity ever true?) (d) Correctly complete the biconditional statement: Let n in N, and let [a] in Zn. Then [a] is a unit in Zn if and only if ...

Units in Zn. Let n be a positive integer. Let [a] in Zn.
(a) Let [a] in Zn with gcd(a, n)=1. Using Bezout's Identity we can find integers x and y so that ax+ny=1. Explain how [a] and [x] are related in Zn.
(b) Prove that if gcd(a, n)=1, then [a] is a unit in Zn.
(c) Prove that if [a] is a unit in Zn, then gcd(a, n)=1. (Hint: Is the converse of Bezout's Identity ever true?)
(d) Correctly complete the biconditional statement:

Let n in N, and let [a] in Zn. Then [a] is a unit in Zn if and only if ...

Transcript text: Units in $\mathbb{Z}_{n}$. Let $n$ be a positive integer. Let $[a] \in \mathbb{Z}_{n}$. (a) Let $[a] \in \mathbb{Z}_{n}$ with $\operatorname{gcd}(a, n)=1$. Using Bezout's Identity we can find integers $x$ and $y$ so that $a x+n y=1$. Explain how $[a]$ and $[x]$ are related in $\mathbb{Z}_{n}$. (b) Prove that if $\operatorname{gcd}(a, n)=1$, then $[a]$ is a unit in $\mathbb{Z}_{n}$. (c) Prove that if $[a]$ is a unit in $\mathbb{Z}_{n}$, then $\operatorname{gcd}(a, n)=1$. (Hint: Is the converse of Bezout's Identity ever true?) (d) Correctly complete the biconditional statement: Let $n \in \mathbb{N}$, and let $[a] \in \mathbb{Z}_{n}$. Then $[a]$ is a unit in $\mathbb{Z}_{n}$ if and only if $\ldots$

Solution

Solution Steps

Solution Approach

(a) In the ring of integers modulo $ n $, $[a]$ and $[x]$ are multiplicative inverses if $\gcd(a, n) = 1$. This means that there exist integers $ x $ and $ y $ such that $ ax + ny = 1 $. In $\mathbb{Z}_n$, this implies that $[a][x] = [1]$, showing that $[x]$ is the multiplicative inverse of $[a]$.

(b) To prove that $[a]$ is a unit in $\mathbb{Z}_n$ when $\gcd(a, n) = 1$, we use the fact that there exist integers $ x $ and $ y $ such that $ ax + ny = 1 $. In $\mathbb{Z}_n$, this simplifies to $[a][x] = [1]$, indicating that $[a]$ has a multiplicative inverse, and thus is a unit.

(c) If $[a]$ is a unit in $\mathbb{Z}_n$, then there exists some $[b]$ such that $[a][b] = [1]$. This implies that $ ab \equiv 1 \pmod{n} $, which means there exist integers $ x $ and $ y $ such that $ ax + ny = 1 $. By Bezout's Identity, this implies $\gcd(a, n) = 1$.

Step 1: Determine if $ a $ is a Unit in $ \mathbb{Z}_n $

To determine if $ a = 3 $ is a unit in $ \mathbb{Z}_7 $, we need to check if $\gcd(3, 7) = 1$. The greatest common divisor of 3 and 7 is indeed 1, which implies that 3 is a unit in $ \mathbb{Z}_7 $.

Step 2: Find the Multiplicative Inverse of $ a $ in $ \mathbb{Z}_n $

Since 3 is a unit in $ \mathbb{Z}_7 $, it has a multiplicative inverse. We use the Extended Euclidean Algorithm to find integers $ x $ and $ y $ such that:

\[ 3x + 7y = 1 \]

Through the algorithm, we find that $ x = 5 $ satisfies this equation, meaning that:

\[ 3 \times 5 \equiv 1 \pmod{7} \]

Thus, the multiplicative inverse of 3 in $ \mathbb{Z}_7 $ is 5.