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It is required to design a binary mod-$5$ synchronous counter using $\text{AB}$ flip-flops such that the output $Q_{2} Q_{1} Q_{0}$ changes as $000 \rightarrow 001 \rightarrow 010 \ldots \ldots$ and so on. The excitation table for the $\mathrm{AB}$ flip-flop is given in the table
$\textbf{Table -11}$
$\begin{array}{||c|c|c|c|c|c|}
\hline A & B & Q_{n+1} \\
 \hline 0 & 0 & 1 \\
0 & 1 & \bar{Q}_{n} \\
1 & 0 & Q_{n} \\
1 & 1 & 0 \\\hline
\end{array}$

  1. Write down the state table for the mod-$5$ counter.
  2. Obtain simplified SOP expressions for the inputs $A_{2}, B_{2}, A_{1}, B_{1}, A_{0}$ and $B_{0}$ in terms of $\mathrm{Q}_{2}$. $\mathrm{Q}_{1}, \mathrm{Q}_{0}$ and their complements.
  3. Hence, complete the circuit diagram for the mod-$5$ counter given in the figure using minimum number of $2$-input NAND-gate only.
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