Number theory LMC and GCF equation.

[mtayab1994]

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.

 

[I like Serena]

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?

 

[mtayab1994]

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?

 

[I like Serena]

Of course 1, but there is also another number.
More importantly, what do Lcm and Gcf have in common?

 

[mtayab1994]

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.

 

[I like Serena]

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?

 

[mtayab1994]

Yes.
Can you say more?

Yes: gcd(x,y) divides lcm(x,y) Right?

 

[I like Serena]

Yep. So...

 

[mtayab1994]

Yep. So...

So 2 divides 5T? I'm not quite sure can you help me a little bit more?

 

[I like Serena]

Uhh... no... that is not true either.

Does T divide the left hand side?

 

[mtayab1994]

Uhh... no... that is not true either.

Does T divide the left hand side?

yes it does so I'm guessing it has to divide -7+2klcm(x,y) right??

 

[I like Serena]

Uhh... yes...?

 

[mtayab1994]

Uhh... yes...?

So... how can i find the values of T then??

 

[mtayab1994]

So the Values of T are those that divide lcm(x,y)??

 

[Joffan]

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)?