Determine if a string containing parentheses is valid. A valid string has matching and properly nested opening and closing parentheses.

Imagine a grid where each cell has a value. Your task is to find the number of island regions where the sum of the values in each island is perfectly divisible by a given number K.

Imagine you're watering a garden using strategically placed taps. Find the fewest taps you need to open to water the entire garden, given each tap's watering range.