One hundred lockers

Idea. A locker ends up open exactly when it was toggled an odd number of times, that is when its number has an odd number of divisors. Only perfect squares do.

Step 1. Person $k$ touches locker $n$ exactly when $k$ divides $n$. So locker $n$ is toggled once per divisor of $n$, and it is open at the end if and only if $n$ has an odd number of divisors.

Step 2. The divisors of $n$ come in pairs $\{d,\frac{n}{d}\}$. A pair has two different members unless $d=\frac{n}{d}$, that is $d=\sqrt{n}$. So the number of divisors is odd exactly when $n$ is a perfect square. For example, 12 has divisors $1,2,3,4,6,12$ (even), while 16 has $1,2,4,8,16$ (odd).

Step 3. The perfect squares up to 100 are $1,4,9,\dots ,100$: there are 10 of them, so 10 lockers are open.

With 1000 lockers the same argument gives 31, since $31^2=961\le 1000<32^2$.