If Ramu and Krishna work on alternate days to complete a work, then the work gets completed in exactly 24 days. If R and K denote the number of days required by Ramu and Krishna respectively to complete the work independently, then how many ordered pairs of integral values of R and K are possible?
If Ramu and Krishna work independently then they are doing work for 12 days. How to find the integral pairs. I am stuck and I think it is a really tricky one
In one day, Ramu can do $\large{\frac{1}{R}}$ of the job.
In one day, Krishna can do $\large{\frac{1}{K}}$ of the job.
In $2$ days, working on alternate days, they can do $\large{\frac{1}{R}+\frac{1}{K}}$ of the job.
In $24$ days, working on alternate days, they can do $12\left(\large{\frac{1}{R}+\frac{1}{K}}\right)$ of the job.
Given that it take $24$ days (assumed exact) for them to complete the job, we get \begin{align*} &12\left(\frac{1}{R}+\frac{1}{K}\right)=1\\[4pt] \implies\;&12(R+K)=RK\\[4pt] \implies\;&RK-12R-12K=0\\[4pt] \implies\;&(R-12)(K-12)=12^2\\[4pt] \end{align*} So for every pair $(a,b)$ of positive integers such that $ab=12^2$, we get a pair $(R,K)=(12+a,12+b)$.
It remains to count the number of positive integer factors of $12^2$.
Since $12^2 = (2^4)(3^2)$, it follows that $12^2$ has $(4+1)(2+1)=15$ positive integer factors, hence there are $15$ valid $(R,K)$ pairs.