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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.
Display the image of the four bags (available as a PowerPoint slide).
Alternatively, you could start with this image, with 7s, 10s, 13s and 16s.
Begin by asking students to explore what happens when they add two, three, four... numbers chosen from a set of bags containing $2$s, $4$s, $6$s and $8$s. Can they explain their findings?
Take Three from Five and What Numbers Can We Make Now? are suitable follow-up problems.