ax + by = cz

where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. The challenge is to either solve that conjecture or come up with a counter-example.

can somebody help me solve this?

Question

ax + by = cz

where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. The challenge is to either solve that conjecture or come up with a counter-example.

can somebody help me solve this?

thomas5267 · Answer

2*10+1*3=1*23
I do not see any common factors since 1 is not a prime.

thomas5267 · Answer

The example I just given may not be very convincing so I will give you another one.

Let c=3 since it is a prime.
Let x and y be positive integers that is greater than 2 and divisible by 3, say x=6 and y=15.
Let a and b be positive integers that are not divisible by 3, say a=2 and b=5.
2*6+5*15=3*z
12+75=3*z
87=3*z
z=29
x=6, y=15, z=29, all greater than 2.
a=2, b=5, c=3, no common prime factors for all three numbers.  In fact, a,b,c are pairwise coprime.