# Division proof and prime

• PennState666
In summary, the conversation discusses determining if p | b given the information that p is a prime number and a and b are positive integers, and that p | a^6 and p | a^3 + b^7. The conversation also mentions the use of prime factorization to determine if p | a and the equations that can be used to show that p | b.

## Homework Statement

(2) P is a prime number and a and b are positive integers .
We Know...
p | a^6 \
and
p | a^3 + b^7.
how do i find out how to prove that p | b?

## Homework Equations

if a | b, then a | bx for every x in Z
if a | b, and a | c, then a | bx + cy for any x,y in Z.

## The Attempt at a Solution

p | (a^3)(a^3)

p | a^6(n) + (a^3 + b^7)m for any m,n in Z.

I can't figure anything out about b :(

Do you know about prime factorization? If p|a^6 then p|a, doesn't it?

we don't know enough about a to be able to say that?

If p is prime, then yes you do.

## 1. What is division proof?

Division proof is a mathematical technique used to prove that one number is divisible by another. It involves dividing the two numbers and showing that there is no remainder, thus proving that the first number is a multiple of the second number.

## 2. How is division proof related to prime numbers?

In division proof, prime numbers play an important role. A prime number is a number that is only divisible by 1 and itself. This means that when using division proof, a prime number can only be evenly divided by 1 and itself, making it a useful tool for determining if a number is divisible by another.

## 3. What is the difference between a prime number and a composite number?

A prime number is a number that is only divisible by 1 and itself, while a composite number is a number that has more than two factors. This means that a composite number can be divided by other numbers besides 1 and itself, making it not a prime number.

## 4. How can division proof be used to find prime numbers?

Division proof can be used to find prime numbers by testing each number to see if it is divisible by any number other than 1 and itself. If it is not divisible by any other number, then it is a prime number. This method is often used to find large prime numbers.

## 5. Can division proof be used to prove the primality of any number?

No, division proof can only be used to prove the primality of a limited number of numbers. It is not a foolproof method and there are some numbers, such as Carmichael numbers, that can pass the division proof test even though they are not prime. Other methods, such as the Sieve of Eratosthenes, are more reliable for finding and proving the primality of numbers.