# Number Theory (2) Homework: Find Integer Orders Modulo 19 & 17

• Shackleford
In summary: So, j = 4. And, a = 2^4 = 16. In summary, the homework statement asks for an integer modulo 19 with orders of 2 and 3, and all integers modulo 17 with an order of 4. The multiplicative order of a modulo n is the smallest integer k > 0 such that ak ≡ 1 (mod n), when gcd(a,n) = 1 and n > 1. The solution for the first question is a = 18 for order 2 and a = 7 for order 3. For the second question, the solution is a = 16, as 2 is a primitive root modulo 17 and ord17(2j
Shackleford

## Homework Statement

1. Find an integer modulo 19 with each of the following orders of 2 and 3.

2. Find all integers modulo 17 such that its order modulo 17 is 4.

## Homework Equations

The multiplicative order of a modulo n, denoted by ordn(a), is the smallest integer k > 0 such that ak ≡ 1 (mod n), when gcd(a,n) = 1 and n > 1.

## The Attempt at a Solution

1) I want to find an integer a such that

a2 ≡ 1 (mod 19), a = 18;

a3 ≡ 1 (mod 19), a = 7.

2) I'm looking at ord17(a) = 4.

Well, two is a primitive root modulo 17, so

ord17(2j) = ord17(2)/gcd(8, j) = 8/ gcd(8, j) which implies that j = 6.

26 ≡ 13 (mod 17), so a = 30 + 17k.

Shackleford said:

## Homework Statement

1. Find an integer modulo 19 with each of the following orders of 2 and 3.

2. Find all integers modulo 17 such that its order modulo 17 is 4.

## Homework Equations

The multiplicative order of a modulo n, denoted by ordn(a), is the smallest integer k > 0 such that ak ≡ 1 (mod n), when gcd(a,n) = 1 and n > 1.

## The Attempt at a Solution

1) I want to find an integer a such that

a2 ≡ 1 (mod 19), a = 18;

a3 ≡ 1 (mod 19), a = 7.

2) I'm looking at ord17(a) = 4.

Well, two is a primitive root modulo 17, so

ord17(2j) = ord17(2)/gcd(8, j) = 8/ gcd(8, j) which implies that j = 6.

26 ≡ 13 (mod 17), so a = 30 + 17k.

I don't think you've actually asked a question. But a=4 works as well, that's not of the form 30+17k. Try to find where you missed that one.

Last edited:
Dick said:
I don't think you've actually asked a question. But a=4 works as well, that not of the form 30+17k. Try to find where you missed that one.

Sorry. I just wanted to make sure that I was construing the questions correctly. I'll take a look.

Ah. I forgot gcd(8, j = 2) = 2.

Last edited:

## 1. What is Number Theory and why is it important?

Number Theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It is important because it has many practical applications in fields such as cryptography, computer science, and physics. It also helps us understand the fundamental nature of numbers and their patterns.

## 2. What is an integer order modulo?

An integer order modulo is a mathematical concept that represents the smallest positive integer that, when multiplied by a given number, gives a remainder of 1 when divided by a specified modulus. In other words, it is the power to which a number must be raised in order to get a remainder of 1 when divided by a certain modulus.

## 3. How do you find integer orders modulo 19 and 17?

To find the integer order modulo 19 or 17 of a given number, you need to first determine all the possible remainders when the number is raised to different powers and divided by the modulus. Then, you need to find the smallest power that gives a remainder of 1. This power is the integer order modulo of the number with respect to the given modulus.

## 4. What are the practical applications of finding integer orders modulo?

Finding integer orders modulo is useful in many practical applications, such as in cryptography for generating secure keys and codes. It is also used in solving problems related to number patterns, divisibility, and modular arithmetic. Additionally, it has applications in computer science for optimizing algorithms and in physics for understanding periodic phenomena.

## 5. Are there any tools or techniques that can help with finding integer orders modulo?

Yes, there are several tools and techniques that can help with finding integer orders modulo. These include using modular arithmetic, number theory theorems, and computer programs or calculators that can perform modular exponentiation. It is also helpful to have a strong understanding of number properties and patterns when working with integer orders modulo.

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