Unlock these lockers

On 1st locker 1st student come.

On 2nd locker 2nd student come.

So on 3rd locker factor of 3 will come. Even factors : Closed Odd factors : Open (numbers with odd factors are perfect) 31 perfect square are there in between 1 to 1000

so 31 open lockers

and 1000-31=969 are closed lockers

Let $\tau \left( n \right)$ be the number of factors of a number $n$ . It's clear that if $\tau \left( n \right)$ is even, the locker will be closed while if it's odd, the locker will be open.

Let ${ p }_{ i }$ be the factors of $n$ . For a non-square number, ${ p }_{ i }$ can be divided in pairs -

${ p }_{ 1 }{ p }_{ \tau \left( n \right) },\quad { p }_{ 2 }{ p }_{ \tau \left( n \right) -1 }\dots$

Such that no numbers in any pair are equal. Hence the, for non-square numbers, $\tau \left( n \right)$ is even.

But in case of square numbers, there is exactly one case $p_a \cdot p_b$ such that $a = b$ . Therefore, the number of factors are odd.

So, we just need to calculate the number of non-square numbers from 1 to 1000. We know that there are 31 square numbers below 1000 because $32^2>1000$ . Hence the number of non-square integers are $1000-31 = 969$ .