I am an 8th grade student, in Algebra IHonors, and I am having a hard time with
ID: 3089603 • Letter: I
Question
I am an 8th grade student, in Algebra IHonors, and I am having a hard time with this problem, and I hopesomeone can help. Thank you in advance to all! When Cramster ;) middleschool opened, the students came up with a plan. the first strudentwalked down the hallway and opened each of the 500 lockers. (Thelockers were number 1 through 500.) The escond studen closed everylocked whose number was divisible by 2. (Locker 4, 6 8, 10, ans soon.) The third student changed every locker whosenumber was divisible by (if the locker was open, it wasclosed, and if it was closed, it was opened.) The fourth studentchanged every locker whose number was divisible by 4. The fifthstudent closed very locker whose number was divisible by 5. Thestudent continued this pattern until all 500 students had taken aturn. When they finished, they noticed some lockers were open, butMOST lockers were closed. Which lockers wereopened? Explain the mathematical reason for those lockers beingopened. I worked to find which lockers were open bynumbering 50 lockers on a whiteboard, and closing/opening them 50times, closing the lockers divisible by the X student I was. Aftergoing through 50 lockers, I noticed the only lockers open were thefollowing: 1, 4, 9, 16, 25, 36, and 49. All perfect squares. Basedon this, I can safely say that this pattern will continue, and allperfect squared lockers will be open. (Please correct me if I amwrong, and tell me why) However, I cannot find the mathematicalreason for only the perfect squares being open. (The underlinedsentence above.) Thank you for any help! I am an 8th grade student, in Algebra IHonors, and I am having a hard time with this problem, and I hopesomeone can help. Thank you in advance to all! When Cramster ;) middleschool opened, the students came up with a plan. the first strudentwalked down the hallway and opened each of the 500 lockers. (Thelockers were number 1 through 500.) The escond studen closed everylocked whose number was divisible by 2. (Locker 4, 6 8, 10, ans soon.) The third student changed every locker whosenumber was divisible by (if the locker was open, it wasclosed, and if it was closed, it was opened.) The fourth studentchanged every locker whose number was divisible by 4. The fifthstudent closed very locker whose number was divisible by 5. Thestudent continued this pattern until all 500 students had taken aturn. When they finished, they noticed some lockers were open, butMOST lockers were closed. Which lockers wereopened? Explain the mathematical reason for those lockers beingopened. I worked to find which lockers were open bynumbering 50 lockers on a whiteboard, and closing/opening them 50times, closing the lockers divisible by the X student I was. Aftergoing through 50 lockers, I noticed the only lockers open were thefollowing: 1, 4, 9, 16, 25, 36, and 49. All perfect squares. Basedon this, I can safely say that this pattern will continue, and allperfect squared lockers will be open. (Please correct me if I amwrong, and tell me why) However, I cannot find the mathematicalreason for only the perfect squares being open. (The underlinedsentence above.) Thank you for any help!Explanation / Answer
The only way for a locker to remain open is for it to have an oddnumber of students open and close it. The perfect numbers arethe only numbers that have an odd number of factors. 9 has1,3,9 and 16 has 1,2,4,8,16. This is because for everyfactor, there is a pair such that the number times its pair equalsthe starting number (for 16, 1 pairs to 16 and 2 pairs to 8). The perfect numbers, however, have a factor with no pairs (itssquare root). For 16, 4 has no pair. For 9, 3 has nopair. Since only the perfect numbers have a factor with nopair, they are the only numbers with an odd number of factors, andthus the only lockers that will remain open.
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