$X$ can do a piece of work in $24$ days, while $Y$ alone can do it in $16$ days. With the help of $Z$ they finish the work in $8$ days. Find in how many days $Z$ can do the work alone?

The total work is the least common multiple of 24 and 16, which is 48 units, making the daily work capacities of X, Y, and their combined effort with Z equal to 2, 3, and 6 units respectively. Since X and Y together complete 5 units daily, Z's daily capacity is $6 - 5 = 1$ unit. Dividing the total work of 48 units by Z's capacity of 1 unit per day shows that Z alone can do the work in 48 days.

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