inverse in Zn? b) Use the definition of multiplicative inverse and trial and error to find the multiplicative inverse of 3 in Z16. Use the multiplicative inverse to solve 3x + 9 = 15 in Z16. c) Use the Pulverizer to find the multiplicative inverse of 37 in Z45 or explain why you cannot do so. If you find a multiplicative inverse, use it to solve 37x + 23 = 11 in Z45. d) Use the Pulverizer to find the multiplicative inverse of 39 in Z45 or explain why you cannot do so. If you find a multiplicative inverse, use it to solve 39x + 10 = 44. e) Use the Pulverizer to find the multiplicative inverse of 15 in Z26 or explain why you cannot do so. If you find a multiplicative inverse, use the multiplicative inverse to solve 15x + 5 = 23 in Z26. f) Use the Pulverizer to find the multiplicative inverse of 18 in Z32 or explain why you cannot do so. If you find a multiplicative inverse, use the multiplicative inverse to solve 18x + 6 = 30 in Z32
T he Correct Answer and Explanation is :
.
a) Definition of a multiplicative inverse in $\mathbb{Z}_n$:
An element $a \in \mathbb{Z}_n$ has a multiplicative inverse if there exists an integer $b \in \mathbb{Z}_n$ such that:
$$
ab \equiv 1 \pmod{n}
$$
This means $b$ is the inverse of $a$ modulo $n$, and $a \cdot b \mod n = 1$.
b) Trial and error to find the inverse of $3$ in $\mathbb{Z}_{16}$:
Try values:
- $3 \cdot 11 = 33 \equiv 1 \pmod{16}$
So, inverse of 3 is 11.
Now solve:
$$
3x + 9 \equiv 15 \pmod{16} \Rightarrow 3x \equiv 6 \pmod{16}
\Rightarrow x \equiv 11 \cdot 6 = 66 \equiv \boxed{2} \pmod{16}
$$
c) Pulverizer for inverse of 37 in $\mathbb{Z}_{45}$:
Use Extended Euclidean Algorithm:
- $\gcd(37, 45) = 1$, so inverse exists.
- Solve $37x \equiv 1 \pmod{45}$
Using Pulverizer:
$$
1 = 7(45) – 8(37) \Rightarrow -8 \cdot 37 \equiv 1 \Rightarrow x \equiv \boxed{37} \pmod{45}
$$
Solve:
$$
37x + 23 \equiv 11 \Rightarrow 37x \equiv -12 \equiv 33
\Rightarrow x \equiv 37 \cdot 33 = 1221 \equiv \boxed{6} \pmod{45}
$$
d) Inverse of 39 mod 45:
$$
\gcd(39, 45) = 3 \neq 1 \Rightarrow \text{No inverse.}
$$
e) Inverse of 15 mod 26:
$$
\gcd(15, 26) = 1 \Rightarrow \text{Inverse exists}
\Rightarrow \boxed{7 \cdot 15 = 105 \equiv 1 \pmod{26}}
$$
Solve:
$$
15x + 5 = 23 \Rightarrow 15x = 18 \Rightarrow x = 7 \cdot 18 = 126 \equiv \boxed{22} \pmod{26}
$$
f) Inverse of 18 mod 32:
$$
\gcd(18, 32) = 2 \neq 1 \Rightarrow \text{No inverse.}
$$
Explanation
The concept of a multiplicative inverse in modular arithmetic is foundational in number theory and cryptography. In the set $\mathbb{Z}_n$, which contains integers modulo $n$, an element $a$ has a multiplicative inverse if there exists some integer $b$ such that $ab \equiv 1 \mod n$. This is only possible when $a$ and $n$ are coprime—that is, $\gcd(a, n) = 1$.
To find an inverse, you can either use trial and error (as in part b), or the Extended Euclidean Algorithm, also called the Pulverizer. The Pulverizer expresses $\gcd(a, n) = 1$ as a linear combination $ax + ny = 1$, from which $x$ gives the inverse of $a$ modulo $n$.
In this set:
- Part b uses trial and error to find the inverse of 3 mod 16, then solves a linear congruence.
- Parts c and e involve using the Pulverizer to find the inverse and then solve equations.
- Parts d and f show cases where the inverse does not exist because the number and the modulus are not coprime.
Understanding when an inverse exists and how to compute it is crucial for solving modular equations and has real-world applications in cryptography (RSA encryption), modular algebra, and computer science algorithms.
