Every positive integer except 1 is a multiple of at least one prime.

[s3a]

Homework Statement

The problem (and its solution) are attached in TheProblemAndSolution.jpg. Specifically, I am referring to problem (c).

Homework Equations

Set theory.
Union.
Integers.
Prime numbers.

The Attempt at a Solution

I see how we have all multiples of all prime numbers in the union of the sets, but I can't see how “every positive integer except 1 is a multiple of at least one prime number”. If I could intuitively grasp the sentence I just quoted, then I can see how to get to the final result.

Any help in getting me to intuitively understand the part I quoted would be greatly appreciated!

 

[mfb]

Every positive integer except 1 is a multiple of at least one prime.

Isn't that a trivial statement given the definition of prime numbers?
Hint: look at the factors of the number.

 

[Dick]

Isn't that a trivial statement given the definition of prime numbers?
Hint: look at the factors of the number.

It's not TOTALLY trivial. If a number $n$ isn't prime then it can be factored into $n=ab$ where neither $a$ nor $b$ is one. If neither one is prime then they are both composite and $a<n$. Then factor $a$. You need to argue that this can't go on forever. It's sort of intuitively obvious but to write a formal proof you need to say that there is not an infinite sequence of decreasing positive integers. There is a little meat here.

 

[s3a]

Thanks for the replies.

I think I now understand it intuitively. Basically, prime numbers are numbers that can't be factored further (except for factors of 1 and the number itself), and all composite numbers can be divided by a prime number, since a prime number is at least one of the multiplicands of the factored form of the composite numbers, right?

Assuming I've understood the intuitive argument, I would like to move to the formal proof, so, Dick or mfb (or anyone else), could you please show me how you would (formally) prove that “every positive integer except 1 is a multiple of at least one prime number”?

 

[mfb]

It's not TOTALLY trivial. If a number $n$ isn't prime then it can be factored into $n=ab$ where neither $a$ nor $b$ is one. If neither one is prime then they are both composite and $a<n$. Then factor $a$. You need to argue that this can't go on forever. It's sort of intuitively obvious but to write a formal proof you need to say that there is not an infinite sequence of decreasing positive integers. There is a little meat here.

I am aware of that, and still consider this as trivial. You can use induction as well.

I think I now understand it intuitively. Basically, prime numbers are numbers that can't be factored further (except for factors of 1 and the number itself), and all composite numbers can be divided by a prime number, since a prime number is at least one of the multiplicands of the factored form of the composite numbers, right?

Right.

Assuming I've understood the intuitive argument, I would like to move to the formal proof, so, Dick or mfb (or anyone else), could you please show me how you would (formally) prove that “every positive integer except 1 is a multiple of at least one prime number”?

Dick gave you one approach, induction is another possible way to prove it.

 

[Dick]

Thanks for the replies.

I think I now understand it intuitively. Basically, prime numbers are numbers that can't be factored further (except for factors of 1 and the number itself), and all composite numbers can be divided by a prime number, since a prime number is at least one of the multiplicands of the factored form of the composite numbers, right?

Assuming I've understood the intuitive argument, I would like to move to the formal proof, so, Dick or mfb (or anyone else), could you please show me how you would (formally) prove that “every positive integer except 1 is a multiple of at least one prime number”?

I agree with mfb. If you think you understand it intuitively, you should start the proof, not us.

 

[s3a]

Well, is it formal to say the following?:

Let any integer greater than 1, referred to by N, be the product of M_1, M_2, . . ., M_i, then N = M_1 * M_2 * . . . * M_i, where each multiplicand of N can be written the product of m_1 * m_2 * . . . * m_j, and so forth until the recursion ends and N is described by the product of numbers that can only be divided by themselves and the number 1, which includes all prime numbers and the number 1, but since multiplying any number by the number 1 doesn't affect the result, we can say that all positive integers greater than 1 can be written as the product of prime numbers.
Q.E.D.

 

[Dick]

Well, is it formal to say the following?:

It's not very good. Use mfb's suggestion and induction. Suppose all integers less than or equal to N have a prime factor. Use that to prove all integers less than or equal to N+1 have a prime factor. Think about it and think about induction.