What is the smallest positive integer $N$ such that $N \div 2$ is a perfect square and $N \div 3$ is a perfect cube?

Step 1 of 3

1. 1.Write $N = 2^a 3^b$; a third prime would need an exponent that is at once even (so $N \div 2$ stays a square) and a multiple of $3$ (so $N \div 3$ stays a cube), hence at least $6$, which makes $N$ larger than $648$.