# Number theory LMC and GCF equation.

1. May 7, 2012

### mtayab1994

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

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

3. 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: May 7, 2012
2. May 7, 2012

### 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?

3. May 7, 2012

### mtayab1994

Is that number 1?

4. May 7, 2012

### I like Serena

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

5. May 7, 2012

### mtayab1994

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. May 7, 2012

### I like Serena

Yes.
Can you say more?

Hmm, that is not true.
Where did the "2" go?

7. May 7, 2012

### mtayab1994

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

8. May 7, 2012

Yep. So...

9. May 7, 2012

### mtayab1994

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

10. May 7, 2012

### I like Serena

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

Does T divide the left hand side?

11. May 7, 2012

### mtayab1994

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

12. May 7, 2012

### I like Serena

Uhh... yes...?

13. May 7, 2012

### mtayab1994

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

14. May 7, 2012

### mtayab1994

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

15. May 7, 2012

### Joffan

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)?