The largest number which always divides the sum of any pair of consecutive odd numbers is

Sum of two consecutive numbers is always divisible by 4, for example

1 + 3 = 4

3 + 5 = 8

5 + 7 = 12

All are divisible by 4.

But 1 + 3 = 4 is neither divisible by 6 nor by 8, therefore 4 is the largest number which always divides the sum of any pair of consecutive odd numbers.

**Additional proof (Not for exam purpose)

Any two consecutive odd numbers will be in the form, 2n – 1 and 2n + 1, for n = 1, 2, 3, …and so on

Hence, their addition

2n – 1 + 2n + 1 = 4n is divisible by 4.