September 2026
In each part of this problem, there is a hallway containing of \(K\) doors numbered consecutively from \(1\) to \(K\) that are all initially closed. To toggle a door means to open it if it is closed and to close it if it is open.
We will also use the notation that for a positive integer \(n\), \(\tau(n)\) is equal to the number of positive integer factors of \(n\). For example, \(\tau(1)=1\) since \(1\) has exactly one positive factor and for a prime number \(p\), we always have \(\tau(p)=2\) since prime numbers have exactly two positive factors. For another example \(\tau(10)=4\) since it has four positive integer factors, \(1\), \(2\), \(5\), and \(10\).
In this part, \(K=100\). \(100\) "steps" are performed as follows:
In step \(1\), every door that is numbered with a multiple of \(1\) is toggled.
In step \(2\), every door that is numbered with a multiple of \(2\) is toggled.
In step \(3\), every door that is numbered with a multiple of \(3\) is toggled.
In step \(n\), every door that is numbered with a multiple of \(n\) is toggled. After all \(100\) steps are performed, which doors are open?
In this part, \(K=100\). As with part (a), \(100\) steps are performed with one step for each integer \(n\) from \(1\) through \(K\). This time, in step \(n\), each door that is numbered with a multiple of \(n\) is toggled \(n\) times. For example, in step \(5\), each door that is numbered with a multiple of \(5\) is to be toggled \(5\) times. After all \(100\) steps are performed, which doors are open?
In this part, \(K=2^9\times3^4\times5^{13}\times7^{12}\). As with parts (a) and (b), a step is performed for each positive integer \(n\) from \(1\) through \(K\). In step \(n\), every door that is numbered by a multiple of \(n\) is toggled \(\tau(n)\) times. For example, in step \(5\), every door that is numbered by a multiple of \(5\) is toggled \(\tau(5)=2\) times.
After all \(K\) steps are performed, is the door numbered with \(K\) open or closed?