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#1 2019-04-22 16:49:58

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?


"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

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#2 2019-04-22 22:16:34

Bob
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Registered: 2010-06-20
Posts: 10,058

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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You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
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#3 2019-04-23 01:49:30

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?


"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

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#4 2019-04-23 02:02:14

Jai Ganesh
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Registered: 2005-06-28
Posts: 45,968

Re: Co-prime numbers

Did it, Amartyanil!


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.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#5 2019-04-29 18:30:47

Alg Num Theory
Member
Registered: 2017-11-24
Posts: 693
Website

Re: Co-prime numbers

Amartyanil wrote:

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.


Me, or the ugly man, whatever (3,3,6)

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#6 2019-06-03 01:58:53

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.


"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

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#7 2019-06-03 15:32:58

Jaspers
Member
Registered: 2019-05-24
Posts: 54

Re: Co-prime numbers

Hi Amartyanil.

Amartyanil wrote:

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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