Thank you for all your correct solutions to Share Bears. Natah from Claremont Fan Court School, Marcus from St Philip's School and Halo from Arden Primary School, Australia were able to find a general pattern. Here is what Halo wrote:

Yasmin and Zach can share a number 2, 4, 6, 8 or 10 bears between themselves. I noticed that all these numbers were even. I think this is because the bears are shared between 2 people, so only multiples of 2 can be shared between them.