Make the greatest and the smallest:
4-digit numbers with the digit 4,1,0 using any one digit twice.

Answer

To solve the problem of creating the greatest and smallest 4-digit numbers using the digits 4, 1, and 0, while allowing one digit to be used twice, we will follow these steps:

### Step 1: Identify the digits
The available digits are 4, 1, and 0. We can use one of these digits twice to form a 4-digit number.

### Step 2: Create the greatest 4-digit number
To create the greatest 4-digit number, we should use the largest digit available (which is 4) twice.

- Using 4 twice gives us the digits: 4, 4, 1, 0.
- Now, we need to arrange these digits in descending order to form the greatest number.

Arranging 4, 4, 1, and 0 in descending order gives us: **4410**.

### Step 3: Create the smallest 4-digit number
To create the smallest 4-digit number, we should use the smallest digit available (which is 0) twice. However, we cannot start a number with 0, so we will use the next smallest digit (which is 1) twice.

- Using 1 twice gives us the digits: 1, 0, 0, 4.
- Now, we need to arrange these digits in ascending order while ensuring that the first digit is not 0.

Arranging 1, 0, 0, and 4 in ascending order while ensuring the first digit is not 0 gives us: **1004**.

### Final Answers
- The greatest 4-digit number is: **4410**
- The smallest 4-digit number is: **1004**

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