# Number Theory - divisibility and primes

• future_phd
In summary, divisibility is the property of a number being able to be divided by another number without leaving a remainder. To determine if a number is divisible by another number, you can use methods such as the division algorithm or the rules of divisibility. A prime number is a positive integer with exactly two distinct divisors, 1 and itself. To determine if a number is prime, you can use methods such as trial division or the Sieve of Eratosthenes. In number theory, prime numbers are considered the building blocks of numbers and have many important applications, such as in cryptography and factorization.
future_phd

## Homework Statement

Prove that any integer n >= 2 such that n divides (n-1)! + 1 is prime.

## The Attempt at a Solution

I'm having trouble getting started, I have no idea how to approach this, can someone give a hint on where to begin maybe because I'm just not seeing it.

Assuming that n divides $$(n-1) \cdot (n-2) \cdots 2 + 1$$. Show that this entails $$(n-1), (n-2), \cdots 2$$ do not divide $$n$$. In other words, nothing less than n divides n (except the trivial case).

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## 1. What is divisibility?

Divisibility is the property of a number being able to be divided by another number without leaving a remainder. For example, 10 is divisible by 2 because when divided, it results in a whole number (5), whereas 10 is not divisible by 3 because when divided, it leaves a remainder of 1.

## 2. How do I determine if a number is divisible by another number?

A number is divisible by another number if it can be divided evenly without leaving a remainder. One way to determine this is by using the division algorithm, which involves dividing the number by the divisor and checking if the remainder is 0. Another method is to use the rules of divisibility, such as the rule for divisibility by 2 (a number is divisible by 2 if its last digit is even).

## 3. What is a prime number?

A prime number is a positive integer that has exactly two distinct divisors, 1 and itself. In other words, a prime number can only be divided by 1 and itself without leaving a remainder. Examples of prime numbers include 2, 3, 5, 7, 11, etc.

## 4. How do I determine if a number is prime?

To determine if a number is prime, you can use a variety of methods such as trial division, which involves testing if the number is divisible by any of the numbers between 2 and its square root. Another method is to use the Sieve of Eratosthenes, which involves creating a list of numbers and crossing out all the multiples of each prime number until you are left with a list of only prime numbers.

## 5. What is the significance of prime numbers in number theory?

Prime numbers are considered the building blocks of numbers in number theory. They have many important applications, such as in cryptography, where they are used to create secure codes. They also play a crucial role in the factorization of numbers and in the study of patterns in numbers.

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