Final answer:
To prove that (a, b) = 1 if and only if (a + b, ab) = 1 for integers a and b, one must use properties of greatest common divisors (GCDs) and show equivalence through logical implications.
Step-by-step explanation:
The question at hand involves proving that for any two integers a, and b, the greatest common divisor (GCD) of a and b is 1 if and only if the GCD of the sum (a + b) and the product (ab) is also 1.
To prove this, we can leverage the fundamental properties of GCDs and their relationships in conjunction with integer combinations.
Starting by assuming that (a, b) = 1, which means that a and b are coprime, we know that there are no common prime factors between a and b.
Following that, it's necessary to demonstrate that any common divisor of a + b and ab must also be a divisor of both a and b, hence it could only be 1.
Conversely, if (a + b, ab) = 1, implying that a + b and ab are coprime, we need to show that a and b cannot have any common divisors other than 1, because if a and b had a common divisor d > 1, then d would divide both a + b and ab, contradicting our assumption that (a + b, ab) = 1.
Therefore, through a series of implications, one can prove the equivalence of these two statements regarding the GCD of integer pairs and their sums and products.