Question

证明：若(f(x),g(x))=1,则,(f(x)g(x),f(x)＋g(x))=1

Answer 1

设(f(x)g(x),f(x)+g(x))=d(x)

所以d(x) | f(x)g(x),d(x) | f(x)+g(x)

因为(f(x),g(x))=1

所以由d(x) | f(x)g(x),得到d(x) | f(x)或d(x) | g(x)

不妨设d(x) | f(x)

由d(x) | f(x)+g(x)得到d(x) | g(x)

所以d(x) | (f(x),g(x))

d(x)=1