Prove that if x,y, and z are integers and

[Bashyboy]

Homework Statement

Prove that if x,y, and z are integers and xyz=1, then x=y=z=1 or two equal -1 and the other is 1.

2. Homework Equations

The Attempt at a Solution

Clearly, if I plug in 1 for each variable, or -1 in for two variables and 1 for the remaining variable, then the equation is satisfied. But how do I know these are the only solutions? This is precisely what I am trying to show. How do I sufficiently demonstrate this?

I thought of writing the equation as $z = \frac{1}{xy}$. But how do I show that, for any choice of integers x and y, besides 1 and -1, z will never be an integer, but a rational number?

 

[haruspex]

What prime factors can x have?

 

[Bashyboy]

I am not sure; I imagine that it would probably be dependent upon certain circumstances, such as the present one, that xyz=1. Is that right?

 

[haruspex]

I am not sure; I imagine that it would probably be dependent upon certain circumstances, such as the present one, that xyz=1. Is that right?

If x has a factor p, what does that equation tell you?

 

[Bashyboy]

That you should be able to divide both sides by p...? I am not certain.

 

[HallsofIvy]

If x> 1 then xyz> yz so yz must be less than 1.
If x< -1 then xyz< yz so yz must be larger than -1.

Of course, none of x, y, or z can be 0.

 

[Bashyboy]

Would there be a way to show that $\frac{1}{xy}$ is always a rational number, for integers x and y?

 

[haruspex]

If x> 1 then xyz> yz

That's not exactly right since yz could be negative.

Would there be a way to show that $\frac{1}{xy}$ is always a rational number, for integers x and y?

As long as x and y are nonzero integers, it clearly is by definition. But how is that useful here? Stick with the factors hint. If a*b = c and p is a factor of a, what can you say about p and c?

 

[Bashyboy]

Well, I was thinking that this would show that there are no other integer solutions, besides the ones we already mentioned.

 

[haruspex]

Well, I was thinking that this would show that there are no other integer solutions, besides the ones we already mentioned.

Proving a number is rational does not mean it is not an integer.

 

[Bashyboy]

Are not integers those rational numbers whose denominator is one? Well, if x and y are not one, then the denominator of 1/xy would never be one, right?

 

[Bashyboy]

To make this a little more precise, I would have to show that the only solutions to $xy = 1$ are 1 and -1. Here is my proof

$xy = 1$

$x^2 y = x$

$x^2 y - x = 0$

$x(y-1) = 0$

This implies that x=0 or y-1 = 0. But x cannot be 0, as this would not satisfy the equation. Hence, it must be true that $y-1 = 0 \implies y=1$. If y=1, then $x \cdot 1 = 1 \implies x =1$. Notice, however, that I do not get the negative one solution. What is wrong with my proof?

 

[vela]

$x^2 y - x \ne x(y-1)$

 

[Bashyboy]

Ah, yes. I see. So, proving xy=1 is, I imagine, similar to proving that xyz=1. Now, supposing I could prove that xy= 1 implies x=y=1 or x=y=-1, would what I said in this post

Are not integers those rational numbers whose denominator is one? Well, if x and y are not one, then the denominator of 1/xy would never be one, right?

still be valid?

 

[haruspex]

Are not integers those rational numbers whose denominator is one? Well, if x and y are not one, then the denominator of 1/xy would never be one, right?

Yes, if you add that step it works, but it is rather a long way round. Do try to answer my question about factors.

 

[Bashyboy]

Would it be that p is a factor of c?

 

[haruspex]

Would it be that p is a factor of c?

Yes. So if c=1 then...?

 

[Bashyboy]

p would have to be one.

 

[haruspex]

p would have to be one.

If positive, yes, but it could also be...?

 

[Bashyboy]

Or negative one.

 

[haruspex]

Or negative one.

Right. Does that give you enough to finish the question?

 

[Bashyboy]

Okay, allow me to summarize things. If $xy=1$ is true, then this basically states that 1 can be factored into two integers $x$ and $y$; and if $x$ or $y$ can be factored into integers, these must also be factors off 1.

Couldn't I just stop at the fact that $xy=1$ indicates that 1 can be factored into the product of two integers, list all of the possible factorizations of 1, whereby I will know what $x$ and $y$ can be?

The only objection that I see is, how do I know that the only factorizations of 1, involving only two integers, are $1 \cdot 1$ and $(-1)(-1)$?

 

[haruspex]

Okay, allow me to summarize things. If $xy=1$ is true, then this basically states that 1 can be factored into two integers $x$ and $y$; and if $x$ or $y$ can be factored into integers, these must also be factors off 1.

Couldn't I just stop at the fact that $xy=1$ indicates that 1 can be factored into the product of two integers, list all of the possible factorizations of 1, whereby I will know what $x$ and $y$ can be?

The only objection that I see is, how do I know that the only factorizations of 1, involving only two integers, are $1 \cdot 1$ and $(-1)(-1)$?

Think about the magnitudes of xy, x and y. This is the hint Halls gave you in post #6.

 

[pasmith]

I thought of writing the equation as $z = \frac{1}{xy}$. But how do I show that, for any choice of integers x and y, besides 1 and -1, z will never be an integer, but a rational number?

Integers are rational numbers.

Consider:

(1) The product of two integers is an integer.
(2) There is no integer $n$ such that $0 < n < 1$.
(3) For every real $x$, if $x > 1$ then $0 < 1/x < 1$.

What can you conclude about $1/|xy|$ if $|xy| > 1$?

 

[Bashyboy]

Integers are rational numbers.

In post #11, I used the fact that integers those rational numbers whose denominator is one. Then I determined for what integers $x$ and $y$ was $xy = 1$, and found that the only solutions were 1 and -1, meaning that, when $x \ne \pm 1$ and $y \ne \pm 1$, $\frac{1}{xy}$ will never be an integer.

Consider:

(1) The product of two integers is an integer.
(2) There is no integer nn such that 0<n<10 < n < 1.
(3) For every real xx, if x>1x > 1 then 0<1/x<10 < 1/x < 1.

What can you conclude about 1/|xy|1/|xy| if |xy|>1|xy| > 1?

If $x$ and $y$ both non-zero integers and both are not $\pm 1$, then $xy$ is some integer and $|xy|$ is the magnitude and is greater than $1$. By result (3), this means that $\frac{1}{|xy|}$ must be within $0$ and $1$. Result (2) states that it is impossible for $\frac{1}{|xy|}$ to be an integer, because there are no integers between $0$ and $1$.