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If x=6m^2+4n^2, and gcd

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Carcass
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If x=6m^2+4n^2, and gcd [#permalink] New post   06 Jul 2021, 07:53
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If \(x=6m^2+4n^2\), and \(gcd(m,n)=2\), what is the greatest even number that must be a factor of x ?


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KarunMendiratta
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Re: If x=6m^2+4n^2, and gcd [#permalink] New post   07 Jul 2021, 02:39
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Carcass wrote:
If \(x=6m^2+4n^2\), and \(gcd(m,n)=2\), what is the greatest even number that must be a factor of x ?


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8


Let, \(m = 2x\) and \(n = 2y\) with \(GCD = 2\)

\(x = 2[3m^2 + 2n^2]\)
\(x = 2[3(2x)^2 + 2(2y)^2]\)
\(x = 2[12x + 8y]\)
\(x = 8[3x + 2y]\)

Hence, greatest even number that must be a factor of \(x\) is \(8\)
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Re: If x=6m^2+4n^2, and gcd [#permalink] New post   16 Aug 2021, 08:48
Why did we write it as 2x and 2y?

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Re: If x=6m^2+4n^2, and gcd [#permalink] New post   19 Aug 2021, 21:40
Viha wrote:
Why did we write it as 2x and 2y?

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2 is gcd of m and n .
that means 2 is a factor of m and n.
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If x=6m^2+4n^2, and gcd [#permalink] New post   20 Aug 2021, 02:22
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Viha wrote:
Why did we write it as 2x and 2y?

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For example:
m = 6 = (2)(3)
Now, n has to be a number which is not a multiple of 3, otherwise the GCD will change

Say, n = 12 = (2)(3)(2)
GCD is 6

Since the Greatest Common Factor is 2
We can write m and n as 2x and 2y where x and y are different numbers with no common factor.
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If x=6m^2+4n^2, and gcd [#permalink]
20 Aug 2021, 02:22
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