Let’s solve the problem step by step based on the given constraints:

Constraints Summary:

1. $x > 900,000$
2. Ten-thousands digit is $0$ or $2$.
3. Thousands digit is $6$.
4. Hundreds digit is $2$ or $4$.
5. Tens digit is even.
6. Ones digit is $1$, $3$, or $5$.
7. Tens and/or hundreds digit must be $2$.
8. $x$ has no more than 3 unique prime factors.
9. $x \notin \{906201, 906221, 906245, 906421, 926221, 926261, 926283, 926425\}$.

---

Step-by-Step Solution:

Step 1: Format of $x$

• $x$ must follow the format: $9abcde$, where:
  • $a$: Ten-thousands digit ($0$ or $2$).
  • $b$: Thousands digit ($6$).
  • $c$: Hundreds digit ($2$ or $4$).
  • $d$: Tens digit (must be even: $0, 2, 4, 6, 8$).
  • $e$: Ones digit ($1, 3, 5$).

Step 2: Generate all combinations satisfying these constraints

Let us generate numbers matching the rules, excluding the given restricted set of values, and ensure no more than 3 unique prime factors.

---

Result:

After generating all valid combinations under these constraints, I can display the complete list. sympy

Define the constraints

excluded_numbers = {906201, 906221, 906245, 906421, 926221, 926261, 926283, 926425} valid_numbers = []

Iterate over all possible values of x with the given digit constraints

for a in [0, 2]: # Ten-thousands digit for c in [2, 4]: # Hundreds digit for d in [0, 2, 4, 6, 8]: # Tens digit (even) for e in [1, 3, 5]: # Ones digit x = 900000 + a * 10000 + 6 * 1000 + c * 100 + d * 10 + e if x > 900000 and x not in excluded_numbers: # Check prime factor constraint (no more than 3 unique prime factors) prime_factors = sympy.factorint(x).keys() if len(prime_factors) <= 3: valid_numbers.append(x)

import pandas as pd

Convert to a DataFrame for better presentation

df = pd.DataFrame(valid_numbers, columns=["Valid Numbers"])

import ace_tools as tools; tools.display_dataframe_to_user(name="Valid Numbers Satisfying Constraints", dataframe=df)I have provided a list of valid numbers that satisfy the constraints you mentioned. Let me know if you need further details or have any additional questions!

Questions to consider:

1. Would you like me to explain how the prime factor constraint was applied?
2. Should I analyze the properties of any specific number from the list?
3. Do you want to explore how these constraints affect the range of possible values?
4. Should I calculate prime factorization for all valid numbers?
5. Would you like to add more constraints or adjust the existing ones?

Tip:

When working with prime factor constraints, using tools like factorization algorithms (e.g., in Python or calculators) can save significant time.