2025/006) Let n be the smallest positive integer such that n is divisible by 20, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$

 Because $n^2$ is a perfect cube so n is a perfect cube

Because $n^3$ is a perfect square so n is a perfect square

so n is a perfect $6^{th}$ power

as n is divisible by 20$2^2 * 5}

so it is of the form $2^x5^y$ where x and y are multiple of 6

so $n = 2^6 * 5^6 = 10^6= 1000000$

number of digits in $n$ = 7