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This problem offers students the opportunity to consider the underlying structure behind multiples and remainders, as well as leading to some very nice generalisations and justifications.
Begin by asking students to explore what happens when they add numbers chosen from a set of bags containing 2s, 4s, 6s and 8s.
They could then consider what happens when they add numbers chosen from a set of bags containing 1s, 11s, 21s and 31s.
Can they explain their findings?
There are a few related problems that students could work on next:
Take Three from Five
Shifting Times Tables
Charlie's Delightful Machine
A Little Light Thinking
Remainders
Where Can We Visit?
Cinema Problem