Number theory LMC and GCF equation.

  • Thread starter mtayab1994
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  • #1
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Homework Statement



(E) : 2Lcm(x,y)-5gcf(x,y0=7

Homework Equations



1- Find the possible values of the the number T=gcf(x,y)

2- Solve in N^2 the equation (E).

The Attempt at a Solution



For number 1 i transformed the equation and I found an equivalence of 2T Ξ 7(mod 5gcf(x,y) is that correct, and how can i proceed.
 
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Answers and Replies

  • #2
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That would not be correct.
It seems you mixed up T and Lcm.

It would be correct that:
5T Ξ -7(mod 2Lcm(x,y))
and also that:
5T Ξ -7(mod 2)
5T Ξ -7(mod Lcm(x,y))

As an extra observation, which number would have to divide the left hand side and therefore also the right hand side?
 
  • #3
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That would not be correct.
It seems you mixed up T and Lcm.

It would be correct that:
5T Ξ -7(mod 2Lcm(x,y))
and also that:
5T Ξ -7(mod 2)
5T Ξ -7(mod Lcm(x,y))

As an extra observation, which number would have to divide the left hand side and therefore also the right hand side?

Is that number 1?
 
  • #4
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Of course 1, but there is also another number.
More importantly, what do Lcm and Gcf have in common?
 
  • #5
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Of course 1, but there is also another number.
More importantly, what do Lcm and Gcf have in common?

LCM*GCF=x*y

and if we make into a form of an equation we get that 5t+7=lcm(x,y)*k where k is a n integer.
 
  • #6
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LCM*GCF=x*y

Yes.
Can you say more?

and if we make into a form of an equation we get that 5t+7=lcm(x,y)*k where k is a n integer.

Hmm, that is not true.
Where did the "2" go?
 
  • #8
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Yep. So...
 
  • #10
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Uhh... no... that is not true either. :confused:

Does T divide the left hand side?
 
  • #11
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Uhh... no... that is not true either. :confused:

Does T divide the left hand side?

yes it does so i'm guessing it has to divide -7+2klcm(x,y) right??
 
  • #12
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Uhh... yes...?
 
  • #14
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So the Values of T are those that divide lcm(x,y)??
 
  • #15
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Yes: gcd(x,y) divides lcm(x,y) Right???

So...
Does gcd(x,y) divide 2*lcm(x,y)?
Does gcd(x,y) divide 5*gcd(x,y)?
Does gcd(x,y) divide 2*lcm(x,y) - 5*gcd(x,y)?
 

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