Question

statistics
finite number system operations
part two: the multiplicative inverse
the multiplicative inverse is also known as the reciprocal.
examples: 2 & 1/2... 5 & 1/5... 2/3 & 3/2... all of the flipped fraction of the number in question.
finite problem: not all numbers have reciprocals.
finite solution: in a finite system, there are no fractions.
lets use 13 as an example. all numbers in mod 13 are integers.
2 & 6 are reciprocals because 2x6 = 12.
rule: all reciprocals in a mod system with a prime mod, include the number one.
this holds for any mod, because 1 times any number equals that number.
in any mod, the multiplicative inverse of a number (x) is whatever number
needs to be multiplied to x to equal one.
example
mod 13 → what is the multiplicative inverse of 4?
this isnt as straightforward - there are 2 parts to this problem.

1. find a multiple of 4 that is congruent to 1 in mod 13.

→ make a list of multiples of 4 until you come across such a number.
→ multiple
14 = 27 = 40
40 is a multiple of 4 that is congruent to 1 in mod 13.
therefore 4 x 10 = 40
therefore 4 x 10 = 1(mod 13)
4 & 10 are inverses
rule: a value, m, can be its own inverse... because 1x1 = 1
rule: 1 will always be its own inverse in any mod.
but
there is a catch. not all multiplicative inverses will exist in mod systems.
rule: m will not have a multiplicative inverse in mod n if m & n share factors.
the mod number will never have a multiplicative inverse
example
multiplicative inverse pairs in mod 11
1 & 1... 2 & 6... 3 & 4... 5 & 9... 7 & 8... 10 & 10... 11 and 11
multiplicative inverse pairs in mod 10
1 & 1... 3 & 7... 9 & 9... 10 & 10
(2, 4, 5, 6, 8 do not have inverses - all share factors with 10)
prove to me youve got this → what is the multiplicative inverse of 4 in mod 9?

Explanation:

Step1: Set up the congruence equation

We want to find \(x\) such that \(4x\equiv1\pmod{9}\). This means \(4x = 9k + 1\) for some integer \(k\), or we need to find an \(x\) and \(k\) that satisfy this equation.

Step2: Check values of \(x\)

We start checking integer values of \(x\) from \(1\).
When \(x = 1\), \(4\times1=4
ot\equiv1\pmod{9}\).
When \(x = 2\), \(4\times2 = 8
ot\equiv1\pmod{9}\).
When \(x= 7\), \(4\times7=28\), and \(28\div9 = 3\) remainder \(1\), so \(28\equiv1\pmod{9}\).

Answer:

7