A particular plant in the garden needs to be watered every 3 days , trimmed every 4 days and fertilised every 8 days . If a gardener performs these 3 tasks on Day 1, list the days that the gardener will carry out at least 2 tasks over a 30-day period .
I'm not sure how to start or understand this question . Can I get a hint .. Thanks for the help ..
Just to generalize the answer, let's look at each of the tasks pairwise:
He waters every $3$ days and he trims every $4$ days, so he does both these tasks every $lcm(3,4) = 12$ days. This means he does both these tasks on days $\{1, 1+12, 1+24\} = \{1, 13, 25\}$.
He waters every $3$ days and he fertilizes every $8$ days, so he does both these tasks every $lcm(3,8) = 24$ days. This means he does both these tasks on days $\{1, 1+24\} = \{1, 25\}$.
He trims every $4$ days and he fertilizes every $8$ days, so he does both these tasks every $lcm(4,8) = 8$ days. This means he does both these tasks on days $\{1, 1+8, 1+16, 1+24\} = \{1, 9, 17, 25\}$.
We can then clearly see that he does two tasks on days that are in the union of these three sets: days $\{1, 9, 13, 17, 25\}$. He does three tasks on days that are in the intersection of these three sets: days $\{1, 25\}$.
Note that
the days that he both waters and trims are of the form $\operatorname{lcm}(3,4)n+1 = 12n+1$
The days that he both waters and fertilizes are of the form $\operatorname{lcm}(3,8)n+1 = 24n+1$
The days that he both trims and fertilizes are of the form $\operatorname{lcm}(4,8)n+1 = 8n+1$
The days that he trims, waters, and fertilizes are of the form $\operatorname{lcm}(3,4,8)n+1 = 24n+1$
day 1: water, trim, fertilize
day 2: nothing
day 4: water
day 5: trim
day 7: water
day 9: trim, ferilize
day 10: water, etc.
you will always trim on days you fertilize. And you will water, trim and fertilize on day 25.
One way to create a list would be to list the days horizontally and the tasks vertically as shown in the following figure:
