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greatest common divisor

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Hello, I have problem with finding every possible number value of [latex]GCD(a^2b+b^2c+c^2a, ab^2+bc^2+ca^2, a+b+c)[/latex] where assumptions are:
[latex]a,b,c[/latex] are three different integer number, [latex]GCD(a, b, c)=1[/latex] and [latex]a,b,c > 1[/latex].


I tried first find the [latex]GCD(a^2b+b^2c+c^2a, ab^2+bc^2+ca^2)[/latex] and next [latex] GCD(GCD(a^2b+b^2c+c^2a, ab^2+bc^2+ca^2), a+b+c)[/latex] by euclidean algorithm but it wasn't work for me, also i don't know how to use the assumption [latex]GCD(a, b, c)=1[/latex]

I checked a couple examples in wolframalpha and it always return 1 so I guess it is the answer but I can't prove it.

Edited by johnmayer18
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