UPSC 2014 CDS I Math Paper

It is given that HCF(a, b) = 1. So, a and b both cannot be even, otherwise their HCF would be 2.
Hence, a and b are either both odd (co-prime) or one is even and other is odd.
When both a and b are odd, then a+b and a–b are even
∴HCF(a+b,a−b)=2
When one is even and other is odd, then a + b and a – b are both odd.
∴HCF(a+b,a−b)=1
Hence, HCF(a+b,a–b) can be either 1 or 2.