Planets X, Y and Z take 360, 450 and 540 days, respectively, to rotate around the same sun. If the three planets are lined up in a ray having the sun as its endpoint, what is the minimum positive number of days before they are all in the exact same locations again?
We are asked to find the least common multiple of 360, 450 and 540.  We prime factorize \begin{align*}
360 &= 2^3\cdot 3^2\cdot 5 \\
450 &= 2 \cdot3^2 \cdot 5^2 \\
540 &= 2^2\cdot 3^3 \cdot 5
\end{align*} and take the largest exponent for each of the primes to get a least common multiple of $2^3\cdot 3^3\cdot 5^2=\boxed{5400}$.