91Ó°ÊÓ

If \(n \in \mathbf{Z}^{+}\)with \(n \geq 4\), verify that \(S(n, n-2)=\left(\begin{array}{l}n \\ 3\end{array}\right)+\) \(3\left(\begin{array}{l}n \\ 4\end{array}\right)\).

For the access function developed in Example \(5.10(\mathrm{~d})\), the matrix \(A=\left(a_{i j}\right)_{m \times n}\) was stored in a one-dimensional array using the row, major implementation. It is also possible to store this matrix using the column major implementation, where each entry \(a_{i i}, 1 \leq i \leq m\), in the first column of \(A\) is stored in locations \(1,2,3, \ldots, m\), respectively, of the array, when \(a_{11}\) is stored in location 1 . Then the entries \(a_{a 2}, 1 \leq i \leq m\), of the second column of \(A\) are stored in locations \(m+1, m+2, m+3, \ldots, 2 m\), respectively, of the array, and so on. Find a formula for the access function \(g\left(a_{i}\right)\) under these conditions.

For \(n \in \mathbf{Z}^{+}\), define \(\tau: \mathbf{Z}^{+} \rightarrow \mathbf{Z}^{+}\)by \(\tau(n)=\) the number of positive-integer divisors of \(n\). a) Let \(n=p_{1}^{\rho_{1}} p_{2}^{2} p_{3}^{e_{3}} \cdots p_{k}^{6 k}\), where \(p_{1}, p_{2}, p_{3}, \ldots, p_{k}\) are distinct primes and \(e_{i}\) is a positive integer for all \(1 \leq i \leq k\). What is \(\tau(n)\) ? b) Determine the three smallest values of \(n \in \mathbf{Z}^{*}\) for which \(\tau(n)=k\), where \(k=2,3,4,5,6\). c) For any \(k \in \mathbf{Z}^{+}, k>1\), prove that \(\tau^{-1}(k)\) is infinite. d) If \(a, b \in \mathbf{Z}^{+}\)with \(\operatorname{gcd}(a, b)=1\), prove that \(\tau(a b)\) \(=\tau(a) \tau(b)\).