P and Q can complete a job in 24 days working together. P can alone complete it in 32 days. Both of them worked together for 8 days and then P left. The number of days Q will take to complete the remaining job is:

P and Q together complete the job in 24 days.

P alone completes the job in 32 days.

P's work rate $= \dfrac{1}{32}$ of the job per day

P and Q together work rate $= \dfrac{1}{24}$ of the job per day

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Q's work rate $= \dfrac{1}{24} - \dfrac{1}{32}$

Finding the common denominator (LCM of 24 and 32 is 96):

$\dfrac{1}{24} = \dfrac{4}{96}$

$\dfrac{1}{32} = \dfrac{3}{96}$

Q's work rate $= \dfrac{4}{96} - \dfrac{3}{96}$

Q's work rate $= \dfrac{1}{96}$ of the job per day

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They worked together for 8 days.

Work completed $= 8 \times \dfrac{1}{24}$

Work completed $= \dfrac{8}{24}$

Work completed $= \dfrac{1}{3}$ of the job

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Remaining work $= 1 - \dfrac{1}{3}$

Remaining work $= \dfrac{2}{3}$ of the job

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Days for Q to complete remaining work $= \dfrac{2}{3} \div \dfrac{1}{96}$

Days $= \dfrac{2}{3} \times 96$

Days $= \dfrac{192}{3}$

Days $= 64$

Therefore, Q will take 64 days to complete the remaining job.