What is the greatest six digit even number that matches all of the clues
The ten thousands digit is twice the tens digit
The hundred thousands digit is more than six
The thousands digit can be divided by the ones digit
No digit is used more than once
5+2x=8
To find the greatest six-digit even number that matches all of the clues, let's examine each clue step-by-step:
1. The ten thousands digit is twice the tens digit.
Let's assume the tens digit is "X". This means the ten thousands digit is "2X".
2. The hundred thousands digit is more than six.
Since no digit is used more than once, the hundred thousands digit must be either 7, 8, or 9. However, the number must be even, so we can eliminate 7 as a possibility. This means the hundred thousands digit is either 8 or 9.
3. The thousands digit can be divided by the ones digit.
Since no digit is used more than once, we need to find a combination of thousands digit and ones digit that can be divided evenly. Let's list all the possibilities:
- Thousands digit: 1, Ones digit: 2 (1 is divisible by 2)
- Thousands digit: 2, Ones digit: 1 (2 is divisible by 1)
- Thousands digit: 3, Ones digit: 6 (3 is divisible by 6)
- Thousands digit: 4, Ones digit: 2 (4 is divisible by 2)
- Thousands digit: 6, Ones digit: 1 (6 is divisible by 1)
- Thousands digit: 8, Ones digit: 4 (8 is divisible by 4)
4. No digit is used more than once.
Since we're looking for the greatest six-digit even number, we want to use the largest digits available. Let's consider all the possibilities:
- 982416 (8 is the greatest available digit for the hundred thousands place, and it's an even number)
- 862914 (8 and 4 are the greatest available digits for the hundred thousands place and thousands place, respectively, and they are both even numbers)
- 862314 (same reasoning as above, with a different arrangement of digits)
Out of these options, the greatest six-digit even number that matches all of the clues is 982416.
Therefore, 982416 is the solution.
To find the greatest six-digit even number that matches all the given clues, we can break down the process step by step.
1. The ten-thousands digit is twice the tens digit: We know that the ten-thousands digit must be even since we are looking for an even number. The possible values for the tens digit are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Let's start with the highest possible value, 9, and work our way down to find the maximum value for the ten-thousands digit.
- If the tens digit is 9, then the ten-thousands digit would be 2 * 9 = 18. However, this violates the condition that no digit should be used more than once.
- If the tens digit is 8, then the ten-thousands digit would be 2 * 8 = 16. This satisfies the condition.
2. The hundred-thousands digit is more than six: Since the ten-thousands digit is already 1, we need the hundred-thousands digit to be higher than 6. The possible values are 7, 8, or 9.
3. The thousands digit can be divided by the ones digit: To satisfy this condition, we need to find a number where the thousands digit is divisible by the ones digit. Since we are looking for the greatest number, we want the thousands digit to be as high as possible, while still meeting the condition.
- If the ones digit is 1, the thousands digit can be any number between 1 and 9.
- If the ones digit is 2, the thousands digit must be an even number (2, 4, 6, or 8).
- If the ones digit is 3, the thousands digit must be a multiple of 3 (3 or 6).
- If the ones digit is 4, the thousands digit must be divisible by 4.
4. No digit is used more than once: We need to ensure that each digit is only used once in our six-digit number.
Based on the conditions, we can try different combinations until we find the greatest number that satisfies all the conditions:
Case 1: ones digit = 1
- thousands digit: 9 (highest value that can be used since 9 is not divisible by any other digit)
- hundred-thousands digit: 7 (higher than 6)
- ten-thousands digit: 16 (twice the tens digit)
- tens digit: 8
- hundreds digit: 5
Result: 987,516
Since this is not an even number, we need to try a different combination.
Case 2: ones digit = 2
- thousands digit: 8 (highest even value that can be used)
- hundred-thousands digit: 9 (higher than 6)
- ten-thousands digit: 18 (twice the tens digit)
- tens digit: 7
- hundreds digit: 5
Result: 987,215
Since this number is an even number and it satisfies all the given clues, we have found the greatest six-digit even number that matches all the conditions.
Therefore, the greatest six-digit even number that matches all the clues is 987,215.
The hundred thousands digit is more than six
XX7XX XX8XX XX9XX
The thousands digit can be divided by the ones digit
and since the number is even, the ones digit is even
X47X2 X87X4
X48X2 X88X2 , but all digits are different, so skip the last, so X48X2
X49X2 X89X4
The ten thousands digit is twice the tens digit
64732 68734 64832 64932 68934
these 5 match your conditions, but the largest is
68934