We will call a number N green if 6 times N contains none of the digits 0, 1, 2, 3, and 4. There are two digits such that every green number contains at least one of them. One of these two digits is 1. What is the other?

To determine which digits \( N \) must contain so that \( 6N \) contains none of the digits 0, 1, 2, 3, and 4, we start by identifying which digits are allowed in \( 6N \). The digits allowed are: 5, 6, 7, 8, and 9.

Given that \( 6N \) must avoid the digits 0, 1, 2, 3, and 4, let's consider \( 6 \times N \) modulo 10 to determine how it affects the possible last digit of \( 6N \).

We analyze the multiplication of \( N \) by 6 for each digit from 0 to 9:
\[
\begin{align*}
6 \times 0 &= 0 & \quad (\text{contains } 0)\\
6 \times 1 &= 6 & \quad (\text{allowed})\\
6 \times 2 &= 12 & \quad (\text{contains } 2)\\
6 \times 3 &= 18 & \quad (\text{contains } 1)\\
6 \times 4 &= 24 & \quad (\text{contains } 2)\\
6 \times 5 &= 30 & \quad (\text{contains } 3)\\
6 \times 6 &= 36 & \quad (\text{contains } 3)\\
6 \times 7 &= 42 & \quad (\text{contains } 2)\\
6 \times 8 &= 48 & \quad (\text{contains } 4)\\
6 \times 9 &= 54 & \quad (\text{allowed})
\end{align*}
\]

From this, the last digits of \( N \) that result in valid \( 6N \) without the digits 0, 1, 2, 3, or 4 are 1 and 9. Thus, \( N \) must end in either 1 or 9 to avoid these digits in \( 6N \).

Next, we check if these constraints (ending in 1 or 9) are enough for \( N \) to be a "green" number. We need to check more digits of \( N \):

If \( N \) ends in 1:
- \( 6N \) will have a form \( 6N = X \cdot 10 + 6 \), so the final digit is 6, which is acceptable.
  
If \( N \) ends in 9:
- \( 6N \) will have a form \( 6N = X \cdot 10 + 54 \), so the final digits are 54, which are both acceptable digits.

Therefore, if \( N \) is "green," it means every "green" number contains at least one of the digits 1 or 9. One of these two digits is given as 1. Therefore, the other digit is:

\[
\boxed{9}
\]

Do you want details or have any questions?

Here are 8 relative questions for you:

1. What is the definition of a "green" number in this context?
2. Can you list the allowed digits that can appear in \( 6N \)?
3. How do we determine the last digit of \( 6N \) from the last digit of \( N \)?
4. What are the constraints on the last digit of \( N \) to avoid digits 0, 1, 2, 3, and 4 in \( 6N \)?
5. Why can't \( N \) end in any digit other than 1 or 9 to be green?
6. How would you verify that 1 and 9 are indeed the only valid digits for green numbers?
7. What mathematical concepts are used in determining green numbers?
8. Can you provide an example of a green number and its calculation?

**Tip:** When solving problems that involve constraints on digits, it's often helpful to check the multiplication results modulo 10 to determine valid digit combinations.

Explore the concept of green numbers in mathematics where a number N is termed green if 6 times N contains none of the digits 0, 1, 2, 3, and 4. Learn about the digits allowed in 6N, how to determine valid last digits of 6N from N, and why numbers ending in 1 or 9 are green. Discover the mathematical principles of number theory and modular arithmetic used to define and analyze green numbers.

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