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Certainly! Let's go through each part of the question step-by-step and determine the answers.
### Part 1: Use the given digits without repetition and make the greatest and smallest 4-digit numbers.
#### (a) Digits: 2, 8, 7, 4
- Greatest 4-digit number: Arrange the digits in descending order: 8, 7, 4, 2. Thus, the greatest number is 8742.
- Smallest 4-digit number: Arrange the digits in ascending order: 2, 4, 7, 8. Thus, the smallest number is 2478.
#### (b) Digits: 9, 7, 4, 1
- Greatest 4-digit number: Arrange the digits in descending order: 9, 7, 4, 1. Thus, the greatest number is 9741.
- Smallest 4-digit number: Arrange the digits in ascending order: 1, 4, 7, 9. Thus, the smallest number is 1479.
#### (c) Digits: 4, 7, 5, 0
- Greatest 4-digit number: Arrange the digits in descending order: 7, 5, 4, 0. Thus, the greatest number is 7540.
- Smallest 4-digit number: Arrange the digits in ascending order: 0, 4, 5, 7. Since 0457 is not a 4-digit number, we need the smallest arrangement of these digits without leading zero: 4, 5, 0, 7. Thus, the smallest number is 4570.
#### (d) Digits: 1, 7, 6, 2
- Greatest 4-digit number: Arrange the digits in descending order: 7, 6, 2, 1. Thus, the greatest number is 7621.
- Smallest 4-digit number: Arrange the digits in ascending order: 1, 2, 6, 7. Thus, the smallest number is 1267.
#### (e) Digits: 5, 4, 0, 3
- Greatest 4-digit number: Arrange the digits in descending order: 5, 4, 3, 0. Thus, the greatest number is 5430.
- Smallest 4-digit number: Arrange the digits in ascending order: 0, 3, 4, 5. Since 0345 is not a 4-digit number, we need the smallest arrangement of these digits without leading zero: 3, 4, 5, 0. Thus, the smallest number is 3450.
### Part 2: Make the greatest and smallest 4-digit numbers by using any one digit twice.
#### (a) Digits: 3, 8, 7
- Greatest 4-digit number: Duplicate the largest digit (8) and arrange in descending order: 8, 8, 7, 3. Thus, the greatest number is 8873.
- Smallest 4-digit number: Duplicate the smallest digit (3) and arrange in ascending order: 3, 3, 7, 8. Thus, the smallest number is 3378.
#### (b) Digits: 9, 0, 5
- Greatest 4-digit number: Duplicate the largest digit (9) and arrange in descending order: 9, 9, 5, 0. Thus, the greatest number is 9950.
- Smallest 4-digit number: Duplicate the smallest digit (0) and arrange in ascending order: 0, 0, 5, 9. Since 0059 is not a 4-digit number, we need the smallest arrangement without leading zero: 0, 5, 9, 0. Thus, the smallest number is 0590. To fix the leading zero issue, rearrange to 590.
#### (c) Digits: 0, 4, 9
- Greatest 4-digit number: Duplicate the largest digit (9) and arrange in descending order: 9, 9, 4, 0. Thus, the greatest number is 9940.
- Smallest 4-digit number: Duplicate the smallest digit (0) and arrange in ascending order: 0, 0, 4, 9. Since 0049 is not a 4-digit number, we need the smallest arrangement without leading zero: 0, 4, 9, 0. Thus, the smallest number is 490.
#### (d) Digits: 8, 5, 1
- Greatest 4-digit number: Duplicate the largest digit (8) and arrange in descending order: 8, 8, 5, 1. Thus, the greatest number is 8851.
- Smallest 4-digit number: Duplicate the smallest digit (1) and arrange in ascending order: 1, 1, 5, 8. Thus, the smallest number is 1158.
These solutions provide the greatest and smallest 4-digit numbers for each set of digits both without repetition and by using one digit twice.
### Part 1: Use the given digits without repetition and make the greatest and smallest 4-digit numbers.
#### (a) Digits: 2, 8, 7, 4
- Greatest 4-digit number: Arrange the digits in descending order: 8, 7, 4, 2. Thus, the greatest number is 8742.
- Smallest 4-digit number: Arrange the digits in ascending order: 2, 4, 7, 8. Thus, the smallest number is 2478.
#### (b) Digits: 9, 7, 4, 1
- Greatest 4-digit number: Arrange the digits in descending order: 9, 7, 4, 1. Thus, the greatest number is 9741.
- Smallest 4-digit number: Arrange the digits in ascending order: 1, 4, 7, 9. Thus, the smallest number is 1479.
#### (c) Digits: 4, 7, 5, 0
- Greatest 4-digit number: Arrange the digits in descending order: 7, 5, 4, 0. Thus, the greatest number is 7540.
- Smallest 4-digit number: Arrange the digits in ascending order: 0, 4, 5, 7. Since 0457 is not a 4-digit number, we need the smallest arrangement of these digits without leading zero: 4, 5, 0, 7. Thus, the smallest number is 4570.
#### (d) Digits: 1, 7, 6, 2
- Greatest 4-digit number: Arrange the digits in descending order: 7, 6, 2, 1. Thus, the greatest number is 7621.
- Smallest 4-digit number: Arrange the digits in ascending order: 1, 2, 6, 7. Thus, the smallest number is 1267.
#### (e) Digits: 5, 4, 0, 3
- Greatest 4-digit number: Arrange the digits in descending order: 5, 4, 3, 0. Thus, the greatest number is 5430.
- Smallest 4-digit number: Arrange the digits in ascending order: 0, 3, 4, 5. Since 0345 is not a 4-digit number, we need the smallest arrangement of these digits without leading zero: 3, 4, 5, 0. Thus, the smallest number is 3450.
### Part 2: Make the greatest and smallest 4-digit numbers by using any one digit twice.
#### (a) Digits: 3, 8, 7
- Greatest 4-digit number: Duplicate the largest digit (8) and arrange in descending order: 8, 8, 7, 3. Thus, the greatest number is 8873.
- Smallest 4-digit number: Duplicate the smallest digit (3) and arrange in ascending order: 3, 3, 7, 8. Thus, the smallest number is 3378.
#### (b) Digits: 9, 0, 5
- Greatest 4-digit number: Duplicate the largest digit (9) and arrange in descending order: 9, 9, 5, 0. Thus, the greatest number is 9950.
- Smallest 4-digit number: Duplicate the smallest digit (0) and arrange in ascending order: 0, 0, 5, 9. Since 0059 is not a 4-digit number, we need the smallest arrangement without leading zero: 0, 5, 9, 0. Thus, the smallest number is 0590. To fix the leading zero issue, rearrange to 590.
#### (c) Digits: 0, 4, 9
- Greatest 4-digit number: Duplicate the largest digit (9) and arrange in descending order: 9, 9, 4, 0. Thus, the greatest number is 9940.
- Smallest 4-digit number: Duplicate the smallest digit (0) and arrange in ascending order: 0, 0, 4, 9. Since 0049 is not a 4-digit number, we need the smallest arrangement without leading zero: 0, 4, 9, 0. Thus, the smallest number is 490.
#### (d) Digits: 8, 5, 1
- Greatest 4-digit number: Duplicate the largest digit (8) and arrange in descending order: 8, 8, 5, 1. Thus, the greatest number is 8851.
- Smallest 4-digit number: Duplicate the smallest digit (1) and arrange in ascending order: 1, 1, 5, 8. Thus, the smallest number is 1158.
These solutions provide the greatest and smallest 4-digit numbers for each set of digits both without repetition and by using one digit twice.
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