The product $$55 \times 60 \times 65$$ is written as the product of five distinct positive integers. What is the least possible value of the largest of these integers?

Here $$55 \times 60 \times 65 = 214500 = 2^2 \cdot 3 \cdot 5^3 \cdot 11 \cdot 13$$. If every factor were at most 19, the primes 11 and 13 would force two of the factors to be exactly 11 and 13, leaving $$\frac{214500}{143} = 1500$$ as a product of three distinct integers at most 19, which is impossible. The largest factor is therefore at least 20, and $$5 \times 11 \times 13 \times 15 \times 20 = 214500$$ shows 20 is attainable.