Form the greatest possible 5-digit number using the clues.All five digits are different. None of the five digit are 1.The digit in the ten thousands place is greater than 7.The sum of all five digit is 18.The greatest digit is equal to the sum of the other four digits.
The digit in the ten thousands place is greater than 7
9xxxx
The sum of all five digit is 18.The greatest digit is equal to the sum of the other four digits.
94320
Let's break down the clues and solve the problem step-by-step:
1. All five digits are different.
This means that none of the digits can be repeated in our 5-digit number.
2. None of the five digits are 1.
This tells us that the digit 1 cannot be used in our 5-digit number.
3. The digit in the ten thousands place is greater than 7.
Since the greatest digit we can use is 9 (since 1 is not allowed), the only possible option for the ten thousands place is 8.
4. The sum of all five digits is 18.
Let's determine the sum of all the digits: 8 + A + B + C + D = 18.
We know that the sum of the four other digits (A, B, C, D) plus 8 equals the greatest digit.
5. The greatest digit is equal to the sum of the other four digits.
From clue 4, we now have: 9 = A + B + C + D.
Let's substitute this value back into clue 4: 8 + A + B + C + D = 18.
This can be rearranged to: A + B + C + D = 10.
Now we have two equations:
1. A + B + C + D = 10
2. A + B + C + D = 9
Since A, B, C, and D must be different digits and the sum cannot exceed 10, there are only three possible combinations:
- A = 1, B = 2, C = 3, D = 4
- A = 2, B = 3, C = 4, D = 1
- A = 3, B = 4, C = 1, D = 2
Let's check which combination satisfies the condition that none of the five digits can be 1:
- If A = 1, we can see that the first combination fails.
Therefore, we have two possible numbers: 82341 and 23412.
However, the digit in the ten thousands place is greater than 7, so the greatest possible 5-digit number using the clues is 82341.
To solve this problem, we need to take into consideration the given clues and use logic to determine the answer step by step.
Let's break down the clues one by one:
1. All five digits are different: This means that no digit can be repeated in the number we are trying to form.
2. None of the digits are 1: This means that the digit 1 cannot be used in our number.
3. The digit in the ten thousands place is greater than 7: Since the digit in this place needs to be greater than 7, it can only be either 8 or 9.
4. The sum of all five digits is 18: This means the sum of each digit in the number we are trying to form should be equal to 18.
5. The greatest digit is equal to the sum of the other four digits: Since the greatest digit is equal to the sum of the other four digits, we can subtract it from the sum of all five digits (which is 18) to find the value of the greatest digit.
Now let's follow these clues to determine the answer:
Step 1: The digit in the ten thousands place is greater than 7, so it can only be 8 or 9.
Step 2: Let's assume that the digit in the ten thousands place is 9. Now, to satisfy the fourth clue which is the sum of all five digits is 18, we can subtract 9 from 18 to get the sum of the remaining four digits, which is 9. This means the sum of the other four digits is 9.
Step 3: We need to find four digits whose sum is 9 and the greatest digit is equal to the sum of the other three digits. The possible combinations could be:
- 9, 3, 2, 1, 1 (But it violates clue 2 as it contains two 1s)
- 8, 3, 2, 2, 2 (But it violates clue 1 as it contains two 2s)
Step 4: Since we cannot find a valid set of four digits for the case when the ten thousands place is 9, let's try the other possibility. Assuming the digit in the ten thousands place is 8, we can follow the same logic to find the value of the greatest digit and four remaining digits whose sum is 9.
Step 5: If the digit in the ten thousands place is 8, then the sum of the remaining four digits should be equal to 18 - 8 = 10. We need to find four digits whose sum is 10 and the greatest digit is equal to the sum of the other three digits.
The only possible combination that satisfies the conditions is:
- 8, 1, 2, 3, 6
So, the greatest possible 5-digit number that satisfies all the given clues is 8,631.
In summary, to solve this problem, we followed the given clues step by step, eliminated possibilities that violated the conditions, and found the valid combination that satisfies all the given conditions, which resulted in the greatest possible 5-digit number being 8,631.