To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote the number of scarfs as `S` and the number of toques as `T`. The profit per scarf is $20, and the profit per toque is $25. Therefore, the total profit can be represented as `20*S + 25*T`.

The constraints are:
1. Time constraint: The factory has 250,000 knitting minutes available. Since a scarf takes 20 minutes to knit and a toque takes 30 minutes to knit, we have `20*S + 30*T <= 250000`.
2. Minimum production requirements: The factory must make at least 5,000 scarfs and at least 3,000 toques. This gives us two constraints: `S >= 5000` and `T >= 3000`.

We want to maximize the total profit under these conditions.

Here is how we can translate this problem into Gurobi code in Python:

```python
from gurobipy import *

# Create a model
m = Model("Knitting_Factory")

# Define decision variables
S = m.addVar(vtype=GRB.INTEGER, name="Scarfs")
T = m.addVar(vtype=GRB.INTEGER, name="Toques")

# Objective function: Maximize profit
m.setObjective(20*S + 25*T, GRB.MAXIMIZE)

# Constraints
m.addConstr(20*S + 30*T <= 250000, "Time_Constraint")
m.addConstr(S >= 5000, "Min_Scarfs")
m.addConstr(T >= 3000, "Min_Toques")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Scarfs: {S.x}")
    print(f"Toques: {T.x}")
    print(f"Maximum Profit: ${20*S.x + 25*T.x}")
else:
    print("No optimal solution found")
```