tag:blogger.com,1999:blog-15628310.post1199758190475791131..comments2021-04-15T02:41:46.574-04:00Comments on Question of the day: Opening Day JittersAnonymoushttp://www.blogger.com/profile/18153935609499338685noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-15628310.post-24836585353417416812008-07-01T10:17:00.000-04:002008-07-01T10:17:00.000-04:00You've got it, Abe. The lockers that remain open ...You've got it, Abe. The lockers that remain open are perfect squares. They are the only numbers divisible by an odd number of whole numbers; every factor other than the number's square root is paired up with another. Thus, these lockers will be "changed" an odd number of times, which means they will be left open. All the other numbers are divisible by an even number of factors and will consequently end up closed.<BR/><BR/><BR/>So the number of open lockers is the number of perfect squares less than or equal to one thousand. These numbers are one squared, two squared, three squared, four squared, and so on, up to thirty one squared. (Thirty two squared is greater than one thousand, and therefore out of range.) So the answer is thirty one.Anonymoushttps://www.blogger.com/profile/18153935609499338685noreply@blogger.comtag:blogger.com,1999:blog-15628310.post-75628096755131424302008-06-30T12:20:00.000-04:002008-06-30T12:20:00.000-04:0031, which is the number of squares between 0 and 1...31, which is the number of squares between 0 and 1000. For a locker number L, where A*B=L, the locker will be open and then shut in pairs (by the Ath and Bth students). In order to be open at the end of the process, the locker number has to have an odd number of factors, and the only way this can happen is if the number is a perfect square (A*A=L).Abehttps://www.blogger.com/profile/04424868492071587450noreply@blogger.com