Complete the following: a. Use each digit once. Make the smallest 5-digit number: b. Use each digit once. Make the largest 5-digit number: c. You can use a digit twice. Make the smallest 5-digit number. d. You can use a digit twice. Make the largest 5-digit number.

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Sure, let's address each part of the problem step by step.

### a. Use each digit once. Make the smallest 5-digit number:

To create the smallest 5-digit number using each digit once, follow these steps:
1. **Choose the smallest non-zero digit for the first position** to ensure it's a 5-digit number. The smallest non-zero digit is **1**.
2. **Use the smallest available digits for the remaining positions**, arranging them in ascending order.

**Result:**
$$ \boxed{10234} $$

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### b. Use each digit once. Make the largest 5-digit number:

To create the largest 5-digit number using each digit once:
1. **Choose the largest available digits** starting from the leftmost position.
2. **Arrange the digits in descending order**.

**Result:**
$$ \boxed{98765} $$

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### c. You can use a digit twice. Make the smallest 5-digit number:

When you are allowed to use one digit twice:
1. **Use the smallest non-zero digit for the first position**. This is **1**.
2. **Use the next smallest digit twice**, which would be **0** (since leading zeros are not allowed, but subsequent zeros are acceptable).
3. **Fill the remaining positions with the next smallest available digits**.

**Result:**
$$ \boxed{10023} $$

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### d. You can use a digit twice. Make the largest 5-digit number:

When you are allowed to use one digit twice:
1. **Use the largest digit twice** to maximize the number. This is **9**.
2. **Fill the remaining positions with the next largest available digits** in descending order.

**Result:**
$$ \boxed{99876} $$

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These are the optimal 5-digit numbers based on the given constraints for each part.
a. The smallest 5-digit number using each digit once is **10234**. b. The largest 5-digit number using each digit once is **98765**. c. The smallest 5-digit number using a digit twice is **10023**. d. The largest 5-digit number using a digit twice is **99876**.

Complete the following: a. Use each digit once. Make the smallest 5-digit number: b. Use each digit once. Make the largest 5-digit number: c. You can use a digit twice. Make the smallest 5-digit number. d. You can use a digit twice. Make the largest 5-digit number.

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