Math Is Fun Forum

Amartyanil

Member

From: Universe

Registered: 2013-05-27

Posts: 82

Co-prime numbers

hi;

If there is an equation satisying the relation:

and

How to prove that

is composite?

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"Every place is the center of the universe. And every moment is the most important moment. And everything is the meaning of life." ~ Dan Harmon

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Co-prime numbers

hi Amartyanil

If you make x the subject:

x = mc/n

We know x is in Z and m,n have no common factors other than 1, so n must divide c, let's say c = dn where d is in Z

So c is the product of two integers, ie is composite.

Bob

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Amartyanil

Member

From: Universe

Registered: 2013-05-27

Posts: 82

Re: Co-prime numbers

hi

I had posted the same thing twice in another thread. Would you please delete it?

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"Every place is the center of the universe. And every moment is the most important moment. And everything is the meaning of life." ~ Dan Harmon

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,425

Re: Co-prime numbers

Did it, Amartyanil!

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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Alg Num Theory

Member

Registered: 2017-11-24

Posts: 693

Website

Re: Co-prime numbers

hi;

If there is an equation satisying the relation:

and

How to prove that

is composite?

That is not true. Take, for example, n = 3, x = 5, m = 5, c = 3. Here, nx = mc and gcd(m,n) = 1 but c=3 is prime.

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Amartyanil

Member

From: Universe

Registered: 2013-05-27

Posts: 82

Re: Co-prime numbers

hi Alg Num Theory;

I should have mentioned that the numbers must be unequal.

Thank you for pointing that out.

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"Every place is the center of the universe. And every moment is the most important moment. And everything is the meaning of life." ~ Dan Harmon

Jaspers

Member

Registered: 2019-05-24

Posts: 54

Re: Co-prime numbers

Hi Amartyanil.

I should have mentioned that the numbers must be unequal.

That is still not enough: the original statement is still untrue. Take

then nx = mc but c = 3 is still prime.

I think the problem needs to be stated this way:

And here is the proof.

If gcd(m,n) = 1, then there are integers r, s such that

Multiplying this by c,

where k = rx+sc. Given that n > 1, c is positive and not equal to n, then k > 1 as well. Thus c = nk is composite.

Last edited by Jaspers (2019-06-03 15:39:33)

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