This problem is a classic example of a pattern involving divisors and the behavior of toggling lockers. Let’s break it down step by step:

Problem Restatement:

• There are 5000 lockers, all initially closed.
• 5000 students pass through the lockers.
• Each student toggles the lockers (opens if closed, closes if open) whose numbers are divisible by their student number.

The question is: Which lockers are left open after all 5000 students have gone through?

Key Observation:

A locker is toggled every time a student number is a divisor of the locker number. For example:

• Locker 12 is toggled by Student 1, Student 2, Student 3, Student 4, Student 6, and Student 12, because those numbers divide 12.

A critical observation is that:

• Each locker is toggled once for every divisor it has.
• If a locker has an even number of divisors, it will end up closed (since it starts closed and every pair of divisors toggles it twice—once open, once closed).
• If a locker has an odd number of divisors, it will end up open (because the final toggle will leave it open).

When does a number have an odd number of divisors?

A number typically has divisors in pairs. For example, 12 has divisors $1 \times 12$, $2 \times 6$, and $3 \times 4$ (all in pairs). But a number will have an odd number of divisors if and only if it is a perfect square. This is because in the case of perfect squares, one divisor is repeated. For example, 36 has divisors $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, and $6 \times 6$. The divisor 6 appears only once (it’s repeated as $6 \times 6$).

Thus, only lockers whose numbers are perfect squares will be left open.

Finding the Perfect Squares:

The lockers left open correspond to the perfect squares between 1 and 5000.

The perfect squares are: $1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, \dots, \lfloor \sqrt{5000} \rfloor^2 = 70^2 = 4900$ Thus, the perfect squares from 1 to 5000 are: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots, 4900$ There are 70 perfect squares (since the largest integer whose square is less than or equal to 5000 is 70).

Conclusion:

The lockers that remain open are those whose numbers are perfect squares. These lockers are: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots, 4900$ Thus, the lockers left open are the lockers numbered by perfect squares between 1 and 5000.

Final Answer:

The lockers that are left open are: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900.$

Would you like further details or have any questions?

Relative Questions:

1. Why do numbers with odd divisors remain open, while those with even divisors close?
2. What is the mathematical property of perfect squares that results in an odd number of divisors?
3. How would the solution change if there were only 100 lockers?
4. Can you prove why non-perfect squares always have an even number of divisors?
5. How can this problem be generalized for any number of lockers and students?

Tip:

When solving problems involving patterns or toggling operations, focus on identifying rules or properties that uniquely impact the final state, such as the number of divisors in this case.