Correct Answer: 21 July 2021

Solution : Given,
​​A and B can individually complete a piece of work in 36 days
⇒ Efficiency of A and B = $\frac{1}{36}$
They started working on 1 July 2021.
A took a day off after every five days of work and B took a day off after every seven days of work.
For 5 days, A and B work together.
⇒ Till 6 July 2021, $\frac{10}{36}$ of the work will be completed
On the 7th day, 7 July 2021, only B worked
⇒ $\frac{10}{36}+\frac{1}{36}$ of the work will be completed.
⇒ $\frac{11}{36}$ of the work will be completed by 7 July 2021.
Again on the 7th day, 8 July 2021, both of them worked together
⇒ $\frac{13}{36}$ of the work will be completed till 8 July, 2021.
Now B is on leave and A alone works.
⇒ $\frac{14}{36}$ of the work will be completed till 9 July, 2021
Now, again both of them will work together for 5 days.
⇒ $\frac{14}{36}+\frac{10}{36}$
⇒ $\frac{24}{36}$ of the work will be completed till 14 July, 2021.
Now, A is going on leave.
⇒ $\frac{25}{36}$ of the work will be completed by 15 July 2021.
Now, for 1 day, A and B again work together.
⇒ $\frac{27}{36}$ of the work will be completed till 16 July, 2021.
Now, B is going on leave and A works alone.
⇒ $\frac{28}{36}$ of the work will be done by 17 July 2021.
Now they work together for 5 days, but they have to work for 4 more days to complete the work
⇒ $\frac{28}{36}+\frac{8}{36}$
⇒ $\frac{36}{36}$ i.e., full work will be completed by 21 July 2021.
Hence, the correct answer is 21 July 2021.