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Logical Puzzles Medium 4 min

There are 100 closed lockers. On pass 1, every locker is opened. On pass 2, every second locker is changed. On pass 3, every third locker is changed, and so on until pass 100. Which lockers remain open?

Hint
A locker changes once for every divisor of its number.

Answer

The lockers numbered 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 remain open.

Explanation

A locker is toggled once for every divisor it has. Most numbers have divisors in pairs, giving an even number of toggles. Perfect squares have one unpaired divisor, their square root, so they are toggled an odd number of times.

Step-by-Step Solution

The open lockers are exactly the perfect squares from 1² through 10².
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