To-Do List App with Sync, Repeating To-Dos
2Do
has all the features that you require (and more):

Due dates ✓
Notes ✓
Sync to a cloud or web-based service where changes can be made ✓ (Dropbox, iCloud and other)
Able to repeat to-dos by due date and last-completed date ✓
Prioritization ✓
Groups / Filters / Folders such as Business, Personal, School ✓
Badge for items due today (tab for today)
Projects (which incorporate other to-do items) ✓

           
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