91Ó°ÊÓ

Compute the addition table and the multiplication table for the integers mod 4 .

Compute the addition table and the multiplication table for the integers mod 5 .

Prove that no two integers in \(Z_{n}\), arithmetic mod \(n\), have the same additive inverse. Conclude from the pigeonhole principle that $$\\{-0,-1,-2, \ldots,-(n-1)\\}=\\{0,1,2, \ldots, n-1\\} .$$

Compute the addition table and multiplication table for the integers mod 6 .

Determine the additive inverses of the integers in \(Z_{8}\), with arithmetic mod 8 .

Determine the additive inverses of \(3,7,8\), and 19 in the integers mod 20 .

For each of the following integers in \(Z_{24}\), determine the multiplicative inverse if a multiplicative inverse exists: $$4,9,11,15,17,23 .$$

Let \(B\) be a BIBD with parameters \(b, v, k, r, \lambda\) whose set of varieties is \(X=\) \(\left\\{x_{1}, x_{2}, \ldots, x_{v}\right\\}\) and whose blocks are \(B_{1}, B_{2}, \ldots, B_{b} .\) For each block \(B_{i}\), let \(\overline{B_{i}}\) denote the set of varieties which do not belong to \(B_{i} .\) Let \(\mathcal{B}^{c}\) be the collection of subsets \(\overline{B_{1}}, \overline{B_{2}}, \ldots, \overline{B_{b}}\) of \(X\). Prove that \(\mathcal{B}^{\mathrm{c}}\) is a block design with parameters $$b^{\prime}=b, v^{\prime}=v, k^{\prime}=v-k, r^{\prime}=b-r, \lambda^{\prime}=b-2 r+\lambda$$ provided that we have \(b-2 r+\lambda>0 .\) The \(\mathrm{BIBD} B^{c}\) is called the complementary design of \(\mathcal{B}\).

Show that \(B=\\{0,1,3,9\\}\) is a difference set in \(Z_{13}\), and use this difference set as a starter block to construct an SBIBD. Identify the parameters of the block design.

Prove that, if we interchange the rows of a Latin square in any way and inter. change the columns in any way, the result is always a Latin square.