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Suppose that x,y,z are positive integers satisfying x is less than or equal to y is less than or equal to z, and such that the product of all three numbers is twice their sum. What is the sum of all possible values of z?

 Sep 18, 2020
 #2
avatar+111127 
+1

Suppose that x,y,z are positive integers satisfying x is less than or equal to y is less than or equal to z, and such that the product of all three numbers is twice their sum. What is the sum of all possible values of z?

 

I just made a possible list and then cancelled the ones that didn't work.   I was left with 3 possible triads

 

You know    \(x\le y\le z\qquad \text{Where that are all positive integers}\)

and you know

 

\(xyz=2(x+y+z)\)

 

the smallest the RHS can be is 6 and the biggest is 54

So the product of x, y and z is between 6 and 54

 

Now I just listed all the possibilities for the triads that multiply to between 6 and 54 and then cancelled them out if they did not fit the equality.

 

Give it a go.

the smallest one is

1,1,6

then

1,1,7

etc.

 

Try doing it yourself.

 Sep 19, 2020
 #3
avatar+401 
+2

Thanks @Melody I will try doing it myself! smiley

 Sep 19, 2020
 #4
avatar+111127 
+1

That is great.

Also, the cube root of 54 is approx 3.8

This means that the smallest one has to be 3 or less.

Melody  Sep 20, 2020

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