# Number theory LMC and GCF equation.

## 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.

Last edited:

## Answers and Replies

I like Serena
Homework Helper
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?

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

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
Homework Helper
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?

Yes.
Can you say more?

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

I like Serena
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Yep. So...

Yep. So...

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

I like Serena
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Uhh... no... that is not true either.

Does T divide the left hand side?

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
Homework Helper
Uhh... yes...?

Uhh... yes...?

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

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

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)?