Chapter 1: Q24E (page 49)

If p is prime, how many elements of ${0, 1, . . . p^{n} - 1}$ have an inverse modulo $p^{n}$ ?

Short Answer

Expert verified

The total number of inverses is $P^{n} - 1 (P - 1) .$ .

Step by step solution

Introduction

The modular inverse of $A \mod C$ is the B value that makes $A * B \mod C = 1$ . The modular multiplicative inverse can be defined as the $GCD (a, b)$ must be equal to 1 and $‘ a ’ and ‘ b ’$ are relatively prime.

Inverse modulo of pn

Given elements are, $\{0, 1, . . . p^{n} - 1\}$ .

If is a prime, then for all given elements i.e. $\{0, 1, . . . p^{n} - 1\}$ , it is not the multiple of p and it contains the inverse of $p^{n}$ .

We have given that the range of elements is $\{0, 1, . . . p^{n} - 1\}$

Now, $p^{n} - 1$ is the only element that get divisible by $p^{n}$ .

So, total number of inverses are,

$P^{n} - P^{n} - 1 = P^{n} - 1 (P - 1)$

Therefore, total number of inverse is $P^{n} - 1 (P - 1)$ .

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If p is prime, how many elements of ${0, 1, . . . p^{n} - 1}$ have an inverse modulo $p^{n}$ ?

Short Answer

Step by step solution

Introduction

Inverse modulo of pn

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If p is prime, how many elements of{0,1,...pn-1} have an inverse modulopn ?

Short Answer

Step by step solution

Introduction

Inverse modulo of pn

One App. One Place for Learning.

Most popular questions from this chapter

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If p is prime, how many elements of ${0, 1, . . . p^{n} - 1}$ have an inverse modulo $p^{n}$ ?