# Math Teacher Help

For the final exam in Math for Elementary Teachers 1, Dr. Vinculum lays out 100 sheets of blank paper, each on one of the desks in the classroom. He has 100 students lined up outside the classroom to take the final. He tells the first student in line to write the numbers 1-100 in order, with 1 number on each sheet of paper. When that student is finished and exits the room, he tells the second student to flip over every paper with an even number on it (the ones numbered 2, 4, 6, 8, etc.). When that student is finished and exits the room, he tells the third student to flip over every third paper (the ones numbered 3, 6, 9, 12, etc.). This student will flip papers that are number side up like three to be number side down, and papers that are number side down like 6 to be number side up. When that student is finished and exits the room, he tells the fourth student to flip over every fourth paper (the ones numbered 4, 8, 12, 16, etc.). This student will also flip some papers from number side up to number side down and other papers from number side down to number side up. He continues to tell students to flip over papers in this pattern (fifth every fifth paper, sixth every sixth paper, etc.) until the hundredth student flips the hundredth paper. When every student has finished, they are thinking they just completed a very easy final exam until Dr. Vinculum asks them to answer the following questions for their final without returning to the room.

Which papers are number-side up?

Explain why these particular papers are the ones that are number-side up?

Suppose Dr. Vinculum had 1,000 students in his class. Which papers would be number-side up?

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