Aditya can complete a work in 24 days alone. Raman can complete the same work in 45 days alone. In how many days can they together complete the work?

$$24=2\times\ 2\times\ 2\times\ 3$$

$$45=3\times\ 3\times\ 5$$

The LCM of 24 and 45 is 360 which can be assumed as the total work.

Aditya can complete a work in 24 days alone.

Efficiency of Aditya = $$\frac{360}{24}$$ = 15 units/day

Raman can complete the same work in 45 days alone.

Efficiency of Raman = $$\frac{360}{45}$$ = 8 units/day

The number of days taken by both of them together to complete the work = $$\frac{360}{15+8}$$

= $$\frac{360}{23}$$ days