Time taken by L alone

Let the daily work rate of L, M, and N be $l, m,$ and $n$ respectively.

$l + m = \frac{1}{72}$ $m + n = \frac{1}{120}$ $l + n = \frac{1}{90}$

Adding all three equations: $2(l + m + n) = \frac{1}{72} + \frac{1}{120} + \frac{1}{90} = \frac{5 + 3 + 4}{360} = \frac{12}{360} = \frac{1}{30}$ $l + m + n = \frac{1}{60}$

Subtracting $(m + n)$ from $(l + m + n)$: $l = \frac{1}{60} - \frac{1}{120} = \frac{1}{120}$

So, L alone can complete the work in 120 days.