We use the fact that a number which is divisible by three primes must be divisible by their product -- this comes from the Fundamental Theorem of Arithmetic. Since we are looking for the least positive integer, we look at the three smallest primes: 2, 3, and 5. Multiplying these yields $2 \times 3 \times 5 = \boxed{30}$, which is the least positive integer divisible by three distinct primes.
What is the least positive integer that is divisible by three distinct primes?
{'answer': '30', 'subject': 'Number Theory', 'level': 2, 'unique_id': 'train/number_theory/484.json'}