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?

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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?

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

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