# Prove Divisibility: $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$

• MHB
• anemone
In summary, the equation that needs to be proven is: $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$. In this context, divisibility means that the left side of the equation is divisible by the right side without leaving a remainder. Proving this equation is important because it is a fundamental concept in number theory and has many applications in various mathematical fields. The steps involved in proving this equation are: expanding the left side of the equation, simplifying the terms, factoring out common factors, and showing that the resulting expression is equal to the right side of the equation. Proving this equation shows that there is a relationship between the three variables
anemone
Gold Member
MHB
POTW Director
Let $x,\,y,\,z$ be integers such that $(x-y)^2+(y-z)^2+(z-x)^2=xyz$, prove that $x^3+y^3+z^3$ is divisible by $x+y+z+6$.

anemone said:
Let $x,\,y,\,z$ be integers such that $(x-y)^2+(y-z)^2+(z-x)^2=xyz$, prove that $x^3+y^3+z^3$ is divisible by $x+y+z+6$.

We know $x^3+y^3+z^3 - 3xyz = \frac{1}{2}(x+y+z)((x-y)^2 + (y-z)^2 + (z-x)^2)$

Hence $x^3+y^3+z^3 = 3xyz + \frac{1}{2}(x+y+z)((x-y)^2 + (y-z)^2 + (z-x)^2)$

Hence $x^3+y^3+z^3 = 3xyz + \frac{1}{2}(x+y+z)(xyz)$ (putting the value from given condition)

Or $x^3+y^3+z^3 = xyz( 3 + \frac{1}{2}(x+y+z))$

Or $x^3+y^3+z^3 = \frac{xyz}{2}( 6 + x+y+z)$

If we can prove that xyz is even then we are through

As (x-y), (y-z) and (z-x) sum to give zero so atleast one of them is even. So xyz is even from the given condition so $\frac{xyz}{2}$ is an integer and hence $x^3+y^3+z^3$ is multiple of $(6 + x+y+z)$

## What is the equation for "Prove Divisibility: $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$?"

The equation is $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$.

## What does "Prove Divisibility" mean in this context?

In this context, "Prove Divisibility" means to show that the equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$ is true for all values of x, y, and z.

## What is the significance of the equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$?

The equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$ is significant because it shows a relationship between the difference of three numbers and their product and sum. It also has applications in number theory and algebraic geometry.

## How can one prove the divisibility of the equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$?

The divisibility of the equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$ can be proven using mathematical induction or by using algebraic manipulation to show that the equation holds true for all values of x, y, and z.

## What are the potential applications of the equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$?

The equation $(x-y)^2+(y-z)^2+(z-x)^2=xyz$ yields $x^3+y^3+z^3$ has potential applications in number theory, algebraic geometry, and cryptography. It can also be used to solve problems involving the sum and product of three numbers.

• General Math
Replies
15
Views
2K
• General Math
Replies
1
Views
836
• General Math
Replies
2
Views
1K
• General Math
Replies
1
Views
926
• General Math
Replies
11
Views
935
• General Math
Replies
9
Views
1K
• General Math
Replies
1
Views
916
• General Math
Replies
4
Views
1K
• General Math
Replies
1
Views
635
• General Math
Replies
2
Views
774