### Instruction: Use each of the digits 3, 4, 6, 8 and 9 exactly once to create the greatest possible five-digit multiple of 6. What is that multiple of 6?### Response: To create the greatest possible five-digit multiple of 6, I need to make the leftmost digits as large as possible and the rightmost digits as small as possible.
I also need to make sure that the number is divisible by 6, which means it must be divisible by both 2 and 3.
To be divisible by 2, the number must end in an even digit, so I have to choose between 4, 6 and 8 for the units digit.
To be divisible by 3, the sum of the digits must be divisible by 3, so I have to keep track of the digit sum as I go.
The largest digit I have is 9, so I want to use it in the ten thousands place.
This gives me a digit sum of 9 so far.
The next largest digit I have is 8, so I want to use it in the thousands place.
This gives me a digit sum of 17 so far.
The next largest digit I have is 6, so I want to use it in the hundreds place.
This gives me a digit sum of 23 so far.
Now I have to choose between 3 and 4 for the tens and units digits.
If I use 4 in the tens place, the digit sum will be 27, which is divisible by 3, but then I have to use 3 in the units place, which is odd and makes the number not divisible by 2.
So I have to use 3 in the tens place and 4 in the units place, which gives me a digit sum of 30, which is divisible by 3, and an even units digit, which makes the number divisible by 2.
Therefore, the greatest possible five-digit multiple of 6 that I can create with the given digits is 98634.