The locker numbers that remain open are the ones which are perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. These numbers have an odd number of factors and remain open after all toggling.
Class 8 Mathematics Ganita Prakash Chapter 1 A Square and A Cube NCERT solutions
Class 8 Mathematics Chapter 1 A Square and A Cube question answer
Only lockers with numbers that are perfect squares remain open because they are toggled an odd number of times. The perfect square numbers from 1 to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Each of these has a repeated middle factor, making their total factor count odd. Hence, these 10 lockers are the only ones that stay open in the end.
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