{ | |
"problem": "What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube?", | |
"level": "Level 3", | |
"type": "Number Theory", | |
"solution": "If $2n$ is a perfect square, then $n$ must be divisible by 2. Now if $3n$ is a perfect cube and $n$ is divisible by 2, then $n$ must be divisible by $3^2=9$ and by $2^3=8$. Therefore, the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube is $9\\times8=\\boxed{72}$." | |
} |