91Ó°ÊÓ

Perform the indicated operations. \(10111_{\text {two }}\) \(\times 1101_{\text {two }}\)

In Exercises \(21-28,\) perform the indicated operations. $$ \begin{array}{r}{10111_{\mathrm{tw} 0}} \\ { \times 1101_{\mathrm{two}}}\end{array} $$

Euler's phi-function \(\varphi\) is another important number-theoretic function on \(\mathbb{N},\) defined by \(\varphi(n)=\) number of positive integers \(\leq n\) and relatively prime to \(n .\) For example, \(\varphi(1)=1=\varphi(\mathbf{2}), \varphi(3)=\mathbf{2}=\varphi(4),\) and \(\varphi(5)=4 .\) Evaluate \(\varphi(n)\) for each value of \(n\). $$15$$

In Exercises \(21-28,\) perform the indicated operations. $$ \begin{array}{r}{1024_ \text { eight }} \\ { \times 2776_ \text { eight }}\end{array} $$

Euler's phi-function \(\varphi\) is another important number-theoretic function on \(\mathbb{N},\) defined by \(\varphi(n)=\) number of positive integers \(\leq n\) and relatively prime to \(n .\) For example, \(\varphi(1)=1=\varphi(\mathbf{2}), \varphi(3)=\mathbf{2}=\varphi(4),\) and \(\varphi(5)=4 .\) Evaluate \(\varphi(n)\) for each value of \(n\). $$17$$

Perform the indicated operations. \(1024_{\text {eight }}\) \(\times 2776_{\text {eight }}\)

In Exercises \(21-28,\) perform the indicated operations. $$ \begin{aligned} & 3 \mathrm{ABC}_{\text { sixteen }} \\ \times & 4 C B A_{\text { sixteen }} \end{aligned} $$

Arrange the binary numbers \(1011,110,11011,10110,\) and 101010 in order of increasing magnitude.

Compute \(\sum_{d | n} \varphi(d)\) for \(n=5,6,10,\) and 12.

A magic square of order \(n\) is a square arrangement of the positive integers 1 through \(n^{2}\) such that the sum of the integers along each row, column, and diagonal is a constant \(k\), called the magic constant. Figure 4.30 shows two magic squares, one of order 3 and the other of order \(4 .\) Prove that the magic constant of a magic square of order \(n\) is \(n\left(n^{2}+1\right) / 2\).