If the sum of two distinct whole numbers is odd, then their difference also must be odd.
Key idea: If a sum is odd, the two addends are of different kinds — one is even and the other is odd.
Let the even number be (2k).
Let the odd number be (2m+1).
Case 1: odd − even
((2m+1) - 2k = 2m - 2k + 1)
(= 2(m-k) + 1)
which is odd.
Case 2: even − odd
(2k - (2m+1) = 2k - 2m - 1)
(= 2(k-m-1) + 1)
which is also odd.
So, no matter which number is larger, the difference of an even and an odd number is odd. Therefore, the statement is true.
Quick check: (8 + 5 = 13) (odd) and (|8 - 5| = 3) (odd).