# Thread: A number theory puzzle

1. ## A number theory puzzle

I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).

2. Originally Posted by wonderboy1953
I give you 7x - 7y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).
Typo? 7*5 - 7*2 most definitely does not equal 1.

In fact it is easily shown that 7x - 7y = 1 has no integer solutions; the LHS is a multiple of 7 while the RHS is not.

3. Originally Posted by undefined
Typo? 7*5 - 7*2 most definitely does not equal 1.

In fact it is easily shown that 7x - 7y = 1 has no integer solutions; the LHS is a multiple of 7 while the RHS is not.

4. Originally Posted by wonderboy1953
I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).
We have Bezout's identity with a sign change. There are infinite solutions, and a solution exists iff x > 4 and $\displaystyle x\equiv 5\pmod{17}$.

For $\displaystyle k \ge 0$ we solve for y

7*(17k + 5) - 17y = 1

7*17k + 35 - 17y = 1

17y = 7*17k + 34

y = 7k + 2

So the solution set is given by (x,y) = (5 + 17k, 2 + 7k) where k ranges over the non-negative integers.

5. That's the solution using parametric equations.

6. Methinks Sir Diophantine just turned over in his grave.

7. Originally Posted by Wilmer
Methinks Sir Diophantine just turned over in his grave.
I realize my solving for y could be shortened realizing that

7(x + 17) - 17(y + 7) = 7x - 17y

but what exactly are you referring to?

8. Originally Posted by Wilmer
Methinks Sir Diophantine just turned over in his grave.
Certainly not English.

9. Hello, wonderboy1953!

I have a very elementary solution . . .

$\displaystyle \text{Given: }\;7x - 17y \:=\: 1$

$\displaystyle \text{How many solutions in natural numbers are there for the equation?}$

Answer: there are a brizillian solutions.

Solve for $\displaystyle x\!:\;\;x \:=\:\dfrac{17y+1}{7} \;=\;2y + \dfrac{3y+1}{7}$

Since $\displaystyle \,x$ is a natural number, $\displaystyle (3y+1)$ must be divisible by 7.

We find that: .$\displaystyle y \;=\;2,\,9,\,16,\,\hdots,\,7t-5\;\text{ for }t \in N$

Then: .$\displaystyle x \;=\;2(7t-5) + \dfrac{3(7t-5)+1}{7} \;=\;17t - 12$

Solutions: .$\displaystyle \begin{Bmatrix}x &=& 17t - 12 \\ y &=& 7t-5 \end{Bmatrix}\;\;\text{ for }t \in N$

Of course, this is basicaly undefined's solution.

10. Originally Posted by undefined
I realize my solving for y could be shortened realizing that
7(x + 17) - 17(y + 7) = 7x - 17y
but what exactly are you referring to?
Nothing in particular UnD; I had just skimmed over:
Diophantine Equations
and saw WB's equation...