I may have misunderstood something, but I believe it is simple to prove that R cannot equal c. Because c is relatively prime to both a and b, there exists some

I may have misunderstood something, but I believe it is simple to prove that R cannot equal c. Because c is relatively prime to both a and b, there exists some prime p which b is divisible by and c is not. Because b is divisible by p, R=rad(a,b,c) is also divisible by p. Since R is divisible by p and c is not, R!=c.