How many distinct positive integers can be represented as the difference of two numbers in the set {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}?
The maximum difference is 18, so the answer will naturally be between 1 and 18.
It is impossible to represent an odd number difference, thus the differences must always be even. Numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, for a total of 9 possibilities.
Proof that you cannot get an odd difference with only odd numbers.
If a is an odd number then it can be written as 2x+1, where x is an integer.
If b is an odd number, then it can be written as 2y+1, where y is an integer.
a-b (odd minus odd) = 2x-1-(2y+1) = 2x-2y. Factoring, we have 2(x-y). Because both x and y are integers, 2(x-y) will always be even, never odd.