use any three of the digits 2,3,4 and 6 to form an 8 digit factor and a two didgit factor with a estimated product that is between 140 and 250

To solve this problem, we need to find three digits out of 2, 3, 4, and 6 that can be used to form an 8-digit factor and a 2-digit factor with a product between 140 and 250.

Let's start by finding the possible 8-digit factor. Since we need eight digits, we will use all four given digits: 2, 3, 4, and 6. To form an 8-digit number, we can use the digits in any order, allowing for repetition.

We can start by exploring different combinations of the four digits. Let's generate all possible combinations using these digits:

1. 2 2 2 2 2 2 2 2
2. 2 2 2 2 2 2 2 3
3. 2 2 2 2 2 2 2 4
4. 2 2 2 2 2 2 2 6
5. 2 2 2 2 2 2 3 2
6. 2 2 2 2 2 2 3 3
7. 2 2 2 2 2 2 3 4
8. 2 2 2 2 2 2 3 6
9. 2 2 2 2 2 2 4 2
10. 2 2 2 2 2 2 4 3
11. 2 2 2 2 2 2 4 4
12. 2 2 2 2 2 2 4 6
13. 2 2 2 2 2 2 6 2
14. 2 2 2 2 2 2 6 3
15. 2 2 2 2 2 2 6 4
16. 2 2 2 2 2 2 6 6

By going through all possible combinations, we can identify the 8-digit factors. However, it seems challenging to do that manually, especially since there are 16 combinations to evaluate. So, let's resort to an easier approach.

To generate these combinations automatically, we can use Python programming language. Here's a Python code snippet that generates all the possible 8-digit numbers using the given digits:

```python
import itertools

digits = [2, 3, 4, 6]
combinations = list(itertools.product(digits, repeat=8))

for combination in combinations:
number = int(''.join(map(str, combination)))
print(number)
```

Now, let's move on to finding a 2-digit factor with an estimated product between 140 and 250. Since we need a 2-digit product, the factors should be 2 to 9. However, we have only three digits available: 2, 3, and 4.

To generate all possible 2-digit factors, we can again use Python. Here's the Python code snippet for that:

```python
factors = []

for i in range(2, 10):
for j in range(2, 10):
factor = i * j
if factor >= 140 and factor <= 250:
factors.append(factor)

for factor in factors:
print(factor)
```

By running the second code snippet, you will find all the 2-digit factors that fall between 140 and 250.

To get the final answer, you need to find the intersection of the 8-digit factor combinations and the 2-digit factors. Therefore, you need to identify if there are any common numbers generated by both code snippets.

I hope this explanation helps you solve the problem!