. For each of the pairs of integers in Exercise 14 , let m...

. For each of the pairs of integers in Exercise 14 , let $m$ denote the first integer and let $n$ denote the second integer of the pair. When it exists, determine the multiplicative inverse of $m$ in $Z_{n}$.

Key Concepts

Modular Arithmetic

Modular arithmetic involves computations where numbers reset or 'wrap around' upon reaching a certain modulus. This branch of arithmetic underpins many areas in number theory and cryptography, focusing on operations within a finite set of residues modulo n.

Multiplicative Inverse in Modular Systems

The multiplicative inverse of an integer a modulo n is an integer x such that the product a · x is congruent to 1 modulo n. This concept is central to solving modular equations and is a critical component in fields like cryptography and computational number theory.

Coprimality and the Greatest Common Divisor

Coprimality between two integers means that they have no common divisors other than 1, which is equivalently stated as their greatest common divisor (GCD) being 1. This property is fundamental in determining whether a multiplicative inverse exists in modular arithmetic.

Extended Euclidean Algorithm

The Extended Euclidean Algorithm is an extension of the standard Euclidean Algorithm that not only computes the greatest common divisor of two integers but also finds integers that express the GCD as a linear combination of the original integers. This method is essential for efficiently finding multiplicative inverses in modular systems.

Transcript

00:01 Hi everyone, so this problem is fairly similar to problem five, so i'm really going to go through the answer.

00:09 So a equals to m equals 17, this is for part a.

00:17 We need to first perform the euclidean algorithm, which is 17 equals to 8 times 2 plus 1.

00:29 Don't forget the remainder.

00:31 I have been there.

00:32 2 equals to 2 times 1.

00:36 So the greatest common division is 1.

00:43 Of a and m is 1.

00:46 So now we need to find greatest common division as a multiple of a and m.

00:56 So we see that greatest common division is equals to 1.

01:01 1 can be simply written as 17 minus 8 times 2 or 1 times 17 minus 7 minus 7 minus 2.

01:16 So our a is 2, so the inverse number is minus 8.

01:24 We have minus 8, mod 17, similar to 9 mod 17.

01:36 So therefore, 9 is also an inverse of 2.

01:44 Move on to b.

01:48 So for b, we have a, sorry, a equals to 34, m equals 89.

02:01 So of course, we performed the euclidean algorithm that 89 equals to 2 times 34 plus 21, 34, equals to 21 plus 13 21 equals to 13 plus 8 13 equals to 8 13 equals to 8 plus 5 8 equals to 5 plus 3 5 equals to 3 plus 2 3 equals to 2 plus 1 and 1 would be our greatest common so again, we find the greatest common division as a multiple of a and m.

02:51 Greater common division is 34 of 3489 equals to 1.

02:58 We need to write it as a number, a number times 34 plus a number times 89 that equals 1...

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