The Solved plugin comes with badges built in for 1, 10, 50, and 150 solutions - you just need to enable them - https://meta.discourse.org/t/discourse-solved/30155#solved-badges-9
Though to create one for 100, you can copy the SQL of one of those, change the threshold, and add it back in as a new custom SQL-triggered badge. There’s more info on that in Create Triggered Custom Badge Queries, and how to enable it here Enable Badge SQL.