Just start by listing the possibilities. If two numbers have the same last digit their difference is divisible by 10 so we may assume all the numbers end in different digits. So the digits could end as:

..0

..1

..2

..3

..4

..5

..6

But 4 + 6 = 10 so the numbers ending in 4 and 6 add to a number divisible by 10. So that's out. The next possibility is:

..0

..1

..2

..3

..4

..5

..7

But the numbers ending in 3 and 7 add to a number divisible by 10, etc.

See if you can generalize this argument and see if you can explain why if you have a list of 6 numbers this theorem isn't true.

-Dan