Final answer:
To prove that (x,y)=(y,z)=1, assume that (x,y) is not 1 and derive a contradiction to show that (x,y) must be 1. Apply the same logic to prove that (y,z) is 1. Hence, the result is proven.
Step-by-step explanation:
To prove that (x,y)=(y,z)=1, we need to show that x and y are coprime (have no common factors other than 1), and y and z are coprime. We can start by assuming that (x,y) is not 1, which means x and y have a common factor. Therefore, let's say there exists a prime number p that divides both x and y. As a result, we can write x = p * x' and y = p * y', where x' and y' are integers. Substituting these values into the equation x² + y² = z², we get (p * x')² + (p * y')² = z², which simplifies to p² * (x'² + y'²) = z². This implies that p divides z², which contradicts the fact that (x,y,z) = 1.
Therefore, (x,y) must be 1. Similarly, we can prove that (y,z) is also 1 using the same logic. Hence, we have successfully shown that (x,y)=(y,z)=1.