This smallest is not that small....

To find the smallest number with the given properties, we note that we only need to have prime factors of $2, 3$ and $5$ . To determine the number of $2$ s required, we note that since $2n$ is a perfect square, we need an odd number of $2$ s. Also, since $3n$ is a perfect cube we need the number of $2$ s to be a multiple of 3. Likewise, for $5n$ to be a perfect fifth power, we need the number of $2$ s to be a multiple of $5$ . The smallest number which satisfies these conditions is $15$ . Therefore, n must have a factor of $2^{15}$ . We continue in the same manner to determine that the number of $3$ s needed must be even, a multiple of $5$ and one less than a multiple of $3$ . The smallest choice is therefore $20$ which gives $n$ a factor of $3^{20}$ . Finally, we find the smallest number of $5$ s needed. There must be a multiple of $6 (2$ and $3)$ of them, and one less than a multiple of $5$ . A quick search tells us that $24$ is the smallest such number. Therefore, $n$ must have a factor of $5^{24}$ . Since $n$ is to be as small as possible we can conclude that these are the only factors of n and therefore $n = 2^{15}3^{20}5^{24} =6810125783203125000000000000000$ .