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For the synchronous counter shown in Fig$.3,$ write the truth table of $Q_{0}, Q_{1}$, and $Q_{2}$ after each pulse, starting from $Q_{0}=Q_{1}=Q_{2}=0$ and determine the counting sequence and also the modulus of the counter.

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28 votes

$$\begin{array}{|ccc|ccc|} \hline \textbf{$Q_0$} & \textbf {$Q_1$} &\textbf {$Q_2$} & \textbf {$Q_{0N}$} & \textbf{$Q_{1N }$}&\textbf{$Q_{2N}$} \\\hline \text{0}& \text{0} & \text{0}  & \text{0} & \text{0} &\text{1}\\\hline \text{0}& \text{0} & \text{1}  & \text{1} & \text{1} &\text{0}\\\hline \text{0}& \text{1} & \text{0}  & \text{1} & \text{0} &\text{0} \\\hline\text{0}& \text{1} & \text{1}  & \text{1} & \text{0} &\text{0} \\\hline\text{1}& \text{0} & \text{0}  & \text{0} & \text{0} &\text{0} \\\hline \text{1}& \text{0} & \text{1}  & \text{0} & \text{1} &\text{0} \\\hline \text{1}& \text{1} & \text{0}  & \text{0} & \text{1} &\text{0} \\\hline \text{1}& \text{1} & \text{1}  & \text{0} & \text{1} &\text{0} \\\hline \end{array}$$
$Q_{0N} = Q_0 \implies J_0 =  Q_1 + Q_2, K_0 = 1$

$Q_{1N} = Q_1 \implies J_1 =   Q_2, K_1 = \bar{Q_0}$

$Q_{2N} = Q_2 \implies J_2 =  \bar{Q_1}.\bar{Q_0}, K_2 = 1$

$$0 - 1 - 6 - 2- 4-0$$

So, MOD $5$ counter.

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${\begin{array}{|c|c|c|}\hline
\bf{Q_0}&    \bf{Q_1}&  \bf{Q_2} \\\hline
0&0&0\\ 0&0&1 \\    1&1&0  \\   0&1&0\\   1&0&0  \\   0&0&0 \\ \hline
\end{array}}$

Counting sequence: $ 0-1-6-2-4$

As there are $5$ different states, so $5$ modulus
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