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Prime or Composite? (3) (Posted on 2023-06-10) Difficulty: 3 of 5
Each of a, b, c, x, y, and z is an integer that satisfy this relationship:
         ayz = bzx = cxy
Can a+b+c+x+y+z be a prime number?

If so, provide an example.
If not, prove it with valid reasoning.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution Solution | Comment 1 of 3
The problem as stated allows the variables to take the value 0.  This gives rise to an entire class of degenerate solutions: a=b=c=0 and any x,y,z which sum to a prime.  So a+b+c+x+y+z can be any prime number.

This is rather unsatisfying and I think that is because the conditions of the problem are incorrect.  Change the opening sentence into "Each of a, b, c, x, y, and z is a positive integer that satisfy this relationship:"

Then lets take the compound equality and divide by xyz: a/x = b/y = c/z
Then these are three equal fractions.  There must be a reduced form, call it m/n.  Now we can say there are three values i,j,k such that: a=i*m, x=i*n, b=j*m, y=j*n, c=k*m, z=k*n.

Then a+b+c+x+y+z = i*m+i*n+j*m+j*n+k*m+k*n = (i+j+k)*(m+n).  Since we are working over positive integers then each of i,j,k,m,n are all at least 1.  
Then the two factors i+j+k and m+n are each at least 3 and 2, respectively.  Then the total must be composite. 
So with the revised opening statement, a+b+c+x+y+z can not be a prime number.

  Posted by Brian Smith on 2023-06-10 10:16:35
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