The basic argument you want works something like this. We can deal with all of the numbers modulo $10$, since divisibility by $10$ is all we care about.
Suppose the numbers somehow defied the condition. Then all of the numbers would have to be different modulo $10$; otherwise, the difference between two numbers that are the same modulo $10$ would be divisible by $10$.
Now how about their sums? There are six possible ways the sum of two numbers might be divisible by $10$:
- They are both equivalent to $0$ (modulo $10$).
- They are equivalent to $1$ and $9$.
- They are equivalent to $2$ and $8$.
- They are equivalent to $3$ and $7$.
- They are equivalent to $4$ and $6$.
- Or, they are both equivalent to $5$.
There are thus six different classes of numbers, and whenever two numbers are in the same class, either their sum or their difference (or both) must be divisible by $10$.
Can you take it from there?