Rahul notices that HCF(n, n + 1) = 1 for every positive integer n he tries.
Which reason correctly explains why this is always true?

If d divides both n and n + 1, then d divides (n + 1) - n = 1, so d = 1. The even/odd remark only rules out the common factor 2 and says nothing about 3, 5 or 7; consecutive integers need not include a prime (8 and 9); and LCM(n, n + 1) = n(n + 1) is a CONSEQUENCE of HCF = 1, not the reason for it.