For integers a and b with HCF(a, b) = 1, what is HCF(a + b, a - b)?

Introduction:
When a and b are coprime, the HCF of a + b and a - b depends on their parity. This is a classic result in number theory relating to sums and differences of coprime integers.

Given Data / Assumptions:

• HCF(a, b) = 1

Concept / Approach:
If one of a or b is even and the other is odd, then a + b and a - b are odd and hence coprime. If both are odd, then each of a + b and a - b is even, and their HCF includes a factor 2 but no higher power of 2 because a and b share no odd common factor.

Step-by-Step Solution:

Verification / Alternative check:
Examples: a = 2, b = 1 gives HCF(3, 1) = 1. a = 5, b = 3 gives HCF(8, 2) = 2.

Why Other Options Are Wrong:
Always 1 or always 2 is not correct across all parities; None of the above and always 0 are invalid here.

Common Pitfalls:
Ignoring parity, or assuming a fixed HCF independent of the parity of a and b.

Final Answer:
Either 1 or 2