Is n your Man?

• as you can see
• 72 is 9* 8 which is 3^2 * 2^3
• so for it to be a perfect square it should be 3^2 * 2^4
• evaluating it is 144 = n^2 , so n = 12
• so the largest positive integer that necessarily divides n is itself
• so the answer is 12

Since $n^2$ is square of an integer, the value of $n^2$ must be a perfect square : $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 ....$
In this list, $144$ is the least number which is divisible by $72$ .
So, if $n^2\ = 144$ , then $n = 12$ .
Answer is $12$ .