1999 AIME Problems/Problem 7
There is a set of 1000 switches, each of which has four positions, called , and . When the position of any switch changes, it is only from to , from to , from to , or from to . Initially each switch is in position . The switches are labeled with the 1000 different integers , where , and take on the values . At step i of a 1000-step process, the -th switch is advanced one step, and so are all the other switches whose labels divide the label on the -th switch. After step 1000 has been completed, how many switches will be in position ?
For each th switch (designated by ), it advances itself only one time at the th step; thereafter, only a switch with larger values will advance the th switch by one step provided divides . Let be the max switch label. To find the divisor multiples in the range of to , we consider the exponents of the number . In general, the divisor-count of must be a multiple of 4 to ensure that a switch is in position A:
We consider the cases where the 3 factors above do not contribute multiples of 4.
- Case of no 2's:
- The switches must be . There are odd integers in to , so we have ways.
- Case of a single 2:
- The switches must be one of or or .
- Since the terms and are three valid choices for the factor above.
- We have ways.
The number of switches in position A is .
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