GATE 2022 EE Q.26: solution and answer

Official answer: 10Source: Official answer key, GATE 2022 EE (IIT Kharagpur)

1. Rule for cascading counters: when counters are cascaded, the second stage advances once for every complete cycle of the first. The modulus of the combination is therefore the product of the individual moduli: $${\text{MOD}}_{\text{total}}=M_1\times M_2$$

2. The MOD 2 counter (a single toggle flip-flop) returns to its start state after $2$ clock pulses and gives one output pulse per cycle. That pulse clocks the MOD 5 counter.

3. So the MOD 5 counter advances one state for every $2$ input clock pulses, and it needs $5$ such advances to return to its start state.

4. Clock pulses needed for the whole chain to return to its initial state: $$2\times 5=10$$

5. The chain therefore has $10$ distinct states, so it is a MOD 10 (decade) counter. The order of the stages does not change this: a MOD 5 stage followed by a MOD 2 stage also gives $5\times 2=10$ states. The 7490 decade counter IC is built this way, from a MOD 2 section and a MOD 5 section.