Finding the Greatest EVEN Factor of X: Solving the GCD of m,n=2

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Discussion Overview

The discussion revolves around finding the greatest even factor of the expression X defined as X=6m²+4n², given that the greatest common divisor (GCD) of m and n is 2. Participants explore various approaches to determine the values of m and n, and how these values influence the factors of X, particularly focusing on even factors.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants suggest starting with the expression X=6m²+4n² and finding pairs (m,n) such that GCD(m,n)=2.
  • One participant proposes that m and n can be expressed as m=2p and n=2q, where p and q are co-prime.
  • Another participant questions the necessity of calculating X to find the greatest even factor, suggesting that understanding the structure of X may suffice.
  • There is a suggestion that the greatest even factor of X could be X itself, while others argue that it may be the highest power of 2 that divides X.
  • Some participants explore specific values for p and q, such as (1,3) and (2,3), to see how they affect the outcome for X.
  • One participant calculates X for specific values and arrives at a conclusion, but others challenge the correctness of the calculations.
  • There is a discussion about whether p and q need to be consecutive or if they can be any co-prime numbers.
  • Several participants express confusion about the problem statement and the relationship between the factors of X and the GCD of m and n.
  • One participant concludes that if they can find a pair (p,q) such that a certain expression is odd, then the largest even factor of X is 8.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of calculating X to determine the greatest even factor. There are multiple competing views regarding the relationship between the values of m and n, the structure of X, and the implications for finding even factors.

Contextual Notes

Participants express uncertainty about the definitions and relationships involved, particularly regarding the values of m and n and their implications for X. There are unresolved questions about the necessity of specific values for p and q and how they relate to the GCD condition.

Who May Find This Useful

This discussion may be of interest to those studying number theory, particularly in relation to GCDs, factors, and expressions involving variables in mathematical contexts.

CharlesLin
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this one got me thinking for a while it starts like this:

X=6m2+4n2

and Greatest Common Divisor(GCD) of (m,n)=2

what is the greatest EVEN number that must be a factor of X

I started this question by thinking what they asked, the gratest number that is a factor of X then I need to calculate X
I know that de GCD of m,n is 2, then I just have to find these two numbers, but the list of two numbers with 2 as a GCD start growing

(2,2)
(2,2)
(2,6)
(2,8)
(2,10)...

Therefore, I would like to know if you have a better way of finding "m" and "n", or how I can be sure that the a pair of those that I found is the correct.
 
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CharlesLin said:
this one got me thinking for a while it starts like this:

X=6m2+4n2

and Greatest Common Divisor(GCD) of (m,n)=2

what is the greatest EVEN number that must be a factor of X

I started this question by thinking what they asked, the gratest number that is a factor of X then I need to calculate X
I know that de GCD of m,n is 2, then I just have to find these two numbers, but the list of two numbers with 2 as a GCD start growing

(2,2)
(2,2)
(2,6)
(2,8)
(2,10)...

Therefore, I would like to know if you have a better way of finding "m" and "n", or how I can be sure that the a pair of those that I found is the correct.

gcd of m and n is 2. so we have m =2p and n = 2q with p and q are co-primes so they do not share a common factor. can you proceed from here
 
Ok following that hint that you gave me,

m=2p, n=2q

I try giving values to p and q. I toke in consideration that p and q are co-primes, in other words, they must consecutive.

p=2, q=3

x=6(16)+4(36)
x= 96+144
x=1050

then I have that 1050 is X but how can I know that the values that I chose for p and q are the appropiate?
If i continue form here knowing that the value of X=1050, I know that the gratest EVEN number that is a factor of 1050 ia 1050 Am I right? then 1050 is my answer!
 
CharlesLin said:
Ok following that hint that you gave me,

m=2p, n=2q

I try giving values to p and q. I toke in consideration that p and q are co-primes, in other words, they must consecutive.

p=2, q=3

x=6(16)+4(36)
x= 96+144
x=1050

then I have that 1050 is X but how can I know that the values that I chose for p and q are the appropiate?
If i continue form here knowing that the value of X=1050, I know that the gratest EVEN number that is a factor of 1050 ia 1050 Am I right? then 1050 is my answer!

you do not need x. you need to know the highest power of 2 that divides x.
further computation of x is incorrect. it is 240.
 
p and q need not be consecutive. That being said, try keeping the factor of 6m2 + 4n2 which is a power of 2 at a minimum. So you have m = 2p and n = 2q with p and q coprime odd numbers.
 
Last edited:
Erm... from the expression of x we can tell that x is even.
Isn't the greatest even factor of x then x itself? (Wondering)
 
Yes! I was thinking "minimum even factor of X".

- - - Updated - - -

Which would, of course, be 2. :o
 
ok, I feel more confused...
what I'm looking is X to be able to answer the question of What's the greatest EVEN number that is a factor of X.

the hint you gave me is that

I shoul find p and q which are co-primes but they aren't consecutive...

my question would be p=1 and q=3 could be a possibility?if so I would have X=168, however someone mentioned that I don't need to find X. I really don't understand why? and how do you know that p and q are co-prime and odd values?
 
If you use the hint provided by kaliprasad in his first post, you then obtain:

$$X=6\left(2p\right)^2+4\left(2q\right)^2=24p^2+16q^2$$

What do you get when you factor that?
 
  • #10
well once you factor that expression...

$$8\left(3{x}^{2}+2{y}^{2}\right)$$

buth then, how do I know what is the answer?
 
  • #11
CharlesLin said:
ok, I feel more confused...
what I'm looking is X to be able to answer the question of What's the greatest EVEN number that is a factor of X.

Can you clarify the problem statement?

The greatest even factor (that is not necessarily prime) of X is X itself.
The greatest even prime factor is $2$.
We currently seem to be heading for the greatest power of $2$ that is a factor of X, but that is not what the problem statement is asking.
 
  • #12
CharlesLin said:
well once you factor that expression...

$$8\left(3{x}^{2}+2{y}^{2}\right)$$

buth then, how do I know what is the answer?

Yes, we get:

$$X=6\left(2p\right)^2+4\left(2q\right)^2=24p^2+16q^2=8\left(3p^2+2q^2\right)$$

Now, if we can find at least one ordered pair $(p,q)$ such that the factor $3p^2+2q^2$ is odd, then what may we conclude?
 
  • #13
so we got to this point...
given this equation

$x=6{m}^{2}+4{n}^{2}$

what is the greatest even number that MUST be a Factor of X?

taking m=2p and n=2q

we have

$x=6{\left(2p\right)}^{2}+4{\left(2q\right)}^{2}$

x=$x=6{\left(2{p}^{2}\right)}+4{\left(4{q}^{2}\right)}$

$x=24{p}^{2}+16{q}^{2}$

$x=8\left(3{p}^{2}+2{q}^{2}\right)$

now I have to look for two numbers p and q that give me an odd number for $3{p}^{2}+2{q}^{2}$ is that right?

I choose p=1, q=3

but I still don't see the answer using (1,3) x=42 is this the answer?
 
  • #14
What I was getting at is that if we can find a $(p,q)$ such that $3p^2+2q^2$ is odd (which you did, and in fact as long as $p$ is odd then $3p^2+2q^2$ will be odd) then we may conclude that the largest even number that will always be a factor of $X$ is $8$.
 
  • #15
MarkFL said:
...the largest even number that will always be a factor of $X$ is $8$.
I think that should be: the largest power of 2 that is a factor of any $X$ is $8$.
 
  • #16
I like Serena said:
I think that should be: the largest power of 2 that is a factor of any $X$ is $8$.

The problem asked for the largest even factor of $X$...which does happen to be a power of 2 given that $\gcd(m,n)=\gcd(6,4)=2$. :)
 
  • #17
thank you very much guys I think I almost got it.

so to find the greatest even number that must be a factor of X I don't need to fin X. Right?

then I don't need to find p and q. But then MarkFL how do you know that (m,n) is (6,4)=2
 
  • #18
CharlesLin said:
so to find the greatest even number that must be a factor of X I don't need to fin X. Right?

$X$ is given as a function of $m$ and $n$ and so it isn't one specific value, but varies according to the definition given.

CharlesLin said:
then I don't need to find p and q. But then MarkFL how do you know that (m,n) is (6,4)=2

We are told that $\gcd(m,n)=2$ and we can compute $\gcd(6,4)=2$ since the prime factorizations of these numbers are:

$$6=2\cdot3$$

$$4=2^2$$

and so we see the largest factor common to both is 2.
 
  • #19
ok I understand,

but how do you know that (n,m) is 6,4 and no other number?
 
  • #20
CharlesLin said:
ok I understand,

but how do you know that (n,m) is 6,4 and no other number?

Sorry, I didn't mean to imply that $m$ and $n$ had any set values, I only meant:

$$\gcd(m,n)=2$$

and

$$\gcd(6,4)=2$$
 

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