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.
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.
Solution
