If m, n, n^' ∈ℤ are such that m and n are relatively prime and m |n n^', then show that m |n^'.

If m, n, n^' ∈ℤ are such that m and n are relatively prime...

Key Concepts

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The special properties of primes are fundamental in number theory because they serve as the building blocks of the integers through prime factorization. One important property of primes, often derived from Euclid's lemma, is that if a prime divides the product of two integers, then it must divide at least one of the factors.

Euclid's Lemma

Euclid's lemma states that if a number divides the product of two integers and is relatively prime to one of them, then it must divide the other. This lemma is a cornerstone of number theory and underpins many proofs, including the uniqueness of prime factorizations.

Greatest Common Divisor and Relatively Prime

The greatest common divisor (gcd) of two integers is the largest integer that divides both numbers. Two numbers are said to be relatively prime if their gcd is 1. This concept is crucial because many results in number theory, such as Euclid's lemma, rely on the fact that relatively prime numbers do not share common factors, enabling the cancellation of factors in divisibility arguments.

Divisibility

Divisibility is the property of an integer dividing another integer without leaving a remainder. It is a fundamental concept in number theory that allows us to describe how numbers can factor into one another, leading to deeper results about the structure of the integers.

Transcript

00:01 If m n and n prime are integers such that m and n are relatively prime and n divides the product n times n prime, then show that n divides n prime.

00:19 To deduce that if p is a prime number that is a positive number greater than one and four with divisors only one and the same number.

00:30 And if p divides a product of two integers then it divides one of them that is if a prime number divides a product of two integers then the prime number must divide one of the numbers so for the first part we're going to use exercise 39 that is m and n relatively prime means or implies that there exist or there are integers numbers u and b such that this linear combination u times n plus b equal 1 you can find that in exercise 39 that is the there exists a linear combination through two integers u and b such that the linear combination is is equal to the greatest common divisor, which in this case is one because the numbers are relatively primes.

02:00 That's one thing we know.

02:02 And from the fact that nm divides n times n prime, we can say that there is an integer k, such that n times n prime is equal to m times k.

02:30 That's the definition of n divides n times n prime.

02:40 So with this, now we can say the following.

02:44 We start with this combination or equation here.

02:48 Um plus n plus vn equal 1.

03:01 Then putting this expression n times n prime equal mk.

03:10 Put in that here before.

03:12 We multiply both sides of this equation by n prime.

03:17 So we get u .m .n.

03:19 Prime plus v .n.

03:21 M prime equal n prime because we have multiplied both sides by n prime and now n times m prime using this is mk so we get um m prime plus b mk is equal to m prime that is we are using the inequality putting here that is we are replacing n times n prime by m times k because they are equal so it get this new equation and from here we can get common factor out m so m prime we are reading from the right to the left is equal to m common factor of u n prime plus bk so we have this equation here and we know that u times m prime plus bk is an integer number because all the terms there are integer so you can say that now u m prime plus bk which is the expression in side parenthesis is an integer because u n prime b k are integers okay and it means that n prime is equal to m times p or p in the integer numbers p is equal to this expression here.

05:39 So this means that m divides n prime.

05:49 That's the definition.

05:51 So we have proved that if m divides the product of two integer numbers, one of those numbers is relatively prime with m, and divides the other number two.

06:06 Or not two, but divides the other number.

06:12 Again, we can say that if n divides the product of two numbers, one of which is relatively prime with m, n divides the other number.

06:22 So now let's go to the second part that is suppose or assume now that p is prime.

06:38 That is a positive integer greater than one.

06:43 And with only divisors, positive integers 1 and the number p, that is the only divisors of p are, and the number 1 and p is greater than 1 so that's a prime number and p divides the product of two integers m and n so we can see that if p is relatively prime with m or p is relatively pried with n we can apply the first part we prove here because if a number an integer number divides the product of all the two numbers and that number is relative prime with one of the factors it divides the other factor so if p is relatively prime with m that is p and n and m are relatively prime then applying the first part p divides n and if p is relatively prime with n applying the first part again p divides m so we can say that if p is relatively prime with m or n can apply the first part of this problem to conclude that p divides m or n that is b divide m or b divides m or b divides n that is divides one of the factors of the product m times n which is what we want to prove in fact then it divides one of the factors okay so that is p relative prime with n or be relative private n so the other possibility is that p is not relative brain with m nor with n so let's suppose that p is or let's say this way p and n are not relatively primes and at the same time p and n are not relatively prime either.

10:36 That's because we have this the third possibility that is that is or we can have three possibilities p is relatively prime with m, p is relatively prime with n or p is relatively is not relatively prime with m nor n.

11:03 The first two possibilities we look at first here in the first part, and we prove that either of the two possibilities allow us to conclude that p device, one of the factors.

11:18 So now we are in the third possibility that is p and m are not relatively prime, and p and n are not relatively prime either.

11:27 But p and m not relatively prime implies that there exists a common factor there must exist a common factor of p and m different from plus or less or plus or minus one remember two numbers are relatively prime if the only factor between for those numbers is one or negative one but there are no other.

12:19 So in this case, if they are not relative prime, there must be a common factor that is different from one or negative one...

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