# What Is the Smallest Integer d Such That a^d Equals 1 Mod n?

• cutesteph
In summary, the problem is to find the smallest positive integer d such that a^d = 1 mod n holds true for any given positive integer a and n. This can be solved using Euler's totient function, but only if a and n are coprime. Otherwise, there is no solution.
cutesteph

## Homework Statement

For any positive integer d and n, find the first integer d such that a^d =1 mod n

## Homework Equations

Euler's totient function phi(n) = # of #s relatively prime to n
Will solve the condition if a and n are coprime but not for the first d

## The Attempt at a Solution

If a and n are not coprime then there is no solution.
I am not sure how to find the smallest d such the condition holds. I know that d will be a factor of Euler's totient function.

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For any positive integer d and n, find the first integer d such that a^d =1 mod n
Is d given or is d to be found ? Is a arbitrary ? Or would the exercise have been something along the lines of
For any positive integer a and n, find the first integer d such that a^d =1 mod n

d is to be found. Essentially I want to find the smallest integer d such that a^d = 1 mod n holds true.
Yes, For any positive integer a and n, find the first integer d such that a^d =1 mod n is how the problem should have been stated.

## 1. What does "a^d = 1 mod n" mean?

This notation is used in modular arithmetic to represent the congruence relationship between the numbers a^d and 1. It means that when a^d is divided by n, the remainder will be 1.

## 2. What is the significance of "a^d = 1 mod n"?

This equation has significance in number theory and cryptography. It is used to solve problems involving large numbers and is the basis for many encryption algorithms.

## 3. How do you solve for "a" in "a^d = 1 mod n"?

To solve for "a", you would need to know the values of d and n. Then, you can use the properties of modular arithmetic to manipulate the equation and find the value of "a".

## 4. Can "a^d = 1 mod n" have multiple solutions?

Yes, it is possible for there to be multiple solutions for "a" in this equation. However, the number of solutions will depend on the values of d and n.

## 5. How is "a^d = 1 mod n" related to Euler's totient function?

Euler's totient function, also known as phi function, calculates the number of positive integers that are relatively prime to a given number. This function is closely related to the equation "a^d = 1 mod n" and is used in solving problems involving modular arithmetic and encryption.

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