A troop of 600 soldiers had provisions for 50 days. After 15 days, 180 soldiers left the troop. For how many more days would the remaining food last for the remaining soldiers?

Initially, 600 soldiers had food for 50 days.
After 15 days, the food remaining for 600 soldiers would be for:

$$50 \text{ days} - 15 \text{ days} = 35 \text{ days}$$

The amount of food remaining is equivalent to 600 soldiers eating for 35 days.

$$\text{Remaining food units} = 600 \text{ soldiers} \times 35 \text{ days} = 21000 \text{ soldier-days}$$

Initial soldiers = 600
Soldiers who left = 180

$$\text{Remaining soldiers} = 600 - 180 = 420 \text{ soldiers}$$

Let $D$ be the number of days the remaining food will last for 420 soldiers.
Using the remaining food units:

$$420 \text{ soldiers} \times D \text{ days} = 21000 \text{ soldier-days}$$ $$D = \frac{21000}{420}$$ $$D = \frac{2100}{42}$$ $$D = 50 \text{ days}$$

The remaining food will last for 50 days for the remaining soldiers.

The final answer is $\text{} 50 \text{ days} \text{}$