- #1

LCSphysicist

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- 162

- Homework Statement
- If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2.

u and v are integers.

- Relevant Equations
- .

If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2.

Where (,) means:

$$ux_1 + vx_2 = 1$$

$$u + v(x_2/x_1) = 1/x_1, u(x_1/x_2) + v = 1/x_2$$

$$u + v = 1/x_1 + 1/x_2 - v x_2/x_1 - u x_1/x_2$$

$$u - v = 1/x_1 - 1/x_2 + u x_1/x_2 - v x_2/x_1$$

Now we can express (u+v,u-v). But i am not sure if it will give us an answer.

I think the key is to know what x

Where (,) means:

$$ux_1 + vx_2 = 1$$

$$u + v(x_2/x_1) = 1/x_1, u(x_1/x_2) + v = 1/x_2$$

$$u + v = 1/x_1 + 1/x_2 - v x_2/x_1 - u x_1/x_2$$

$$u - v = 1/x_1 - 1/x_2 + u x_1/x_2 - v x_2/x_1$$

Now we can express (u+v,u-v). But i am not sure if it will give us an answer.

I think the key is to know what x

_{1}and x_{2}are, knowing that u and v are coprimes. But i don't know how to find it.
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