Listing multiples is a good way to solve this question.
Shared among 5 friends equally :
5,10,15,20,25,30,35,40,45,...
Add the 4 remaining plums :
9,14,19,24,29,34,39,44,49,...
Shared among 9 friends equally :
9,18,27,36,45,54,63,72,81,...
Add the 8 remaining plums :
17,26,35,44,53,62,71,80,89,...
As we can see, 44 is the first common number between the two rows.
So least number of plums she has is 44.
What else can we deduce?
5 × 8 = 40, 40 + 4 = 44
In the first case, each friend got 8 plums.
9 × 4 = 36, 36 + 8 = 44
In the second case, each friend got 4 plums.