Counting divisors and the hundred lockers

1. The map $d\mapsto n/d$ is an involution on the set of divisors of $n$. Its fixed points satisfy $d^2=n$. So divisors come in pairs $\{d,n/d\}$ with $d\ne n/d$, except when $n$ is a square, where $\sqrt n$ is unpaired. Hence $d(n)$ is odd iff $n$ is a square.

2. Locker $m$ is touched once by every student $k$ dividing $m$, i.e. $d(m)$ times. It ends open iff $d(m)$ is odd iff $m$ is a perfect square. The open lockers are $1,4,9,16,25,36,49,64,81,100$ (ten lockers).

3. If $n=\prod p_i^{a_i}$ then $d(n)=\prod(a_i+1)$. To get $12$ divisors, the exponents are $\{11\}$, $\{5,1\}$, $\{3,2\}$ or $\{2,1,1\}$. Minimal numbers: $2^{11}=2048$, $2^5\cdot3=96$, $2^3\cdot3^2=72$, $2^2\cdot3\cdot5=60$. The smallest is $60$.

Lesson. Pair divisors $d\leftrightarrow n/d$; and recall $d(n)=\prod(a_i+1)$ to optimise over factorisation types.