Make the greatest and the smallest 5-digit numbers using different digits in which 5 appears at ten's place.

Answer

To solve the problem of making the greatest and smallest 5-digit numbers using different digits with 5 in the ten's place, we will follow these steps:

### Step 1: Understand the requirements
We need to create two 5-digit numbers:
1. The greatest number
2. The smallest number

Both numbers must have the digit '5' in the ten's place, and all digits must be different.

### Step 2: Create the greatest 5-digit number
To create the greatest number, we want the largest digits in the highest places. Since '5' must be in the ten's place, we will place the largest digits in the other places.

- The largest digits available are 9, 8, 7, 6, and 5.
- We will place '5' in the ten's place, so we need to arrange the remaining digits (9, 8, 7, 6) in descending order.

Thus, the arrangement will be:
- Thousands place: 9
- Hundreds place: 8
- Ten's place: 5
- Units place: 7
- Remaining digit: 6

So, the greatest 5-digit number is **98765**.

### Step 3: Create the smallest 5-digit number
To create the smallest number, we want the smallest digits in the highest places. Again, '5' must be in the ten's place.

- The smallest digits we can use are 0, 1, 2, 3, 4, and 5.
- We cannot start with '0' as it would not make it a 5-digit number.
- We will place '5' in the ten's place and arrange the smallest remaining digits (0, 1, 2, 3, 4) in ascending order.

Thus, the arrangement will be:
- Thousands place: 1 (the smallest non-zero digit)
- Hundreds place: 0
- Ten's place: 5
- Units place: 2
- Remaining digit: 3

So, the smallest 5-digit number is **10253**.

### Final Answers:
- The greatest 5-digit number is **98765**.
- The smallest 5-digit number is **10253**.

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