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?
A multiple of 6 is a multiple of 3, which means the sum of its digits is a multiple of 3, and a multiple of 2, which means its units digit is even. We note that 4, 6, and 8 are even and that $3+4+6+8+9=30=3(10)$, so indeed it is possible to create a multiple of 6. Choose the smallest even number on the list, 4, for the units digit and arrange the rest of the digits in decreasing order to maximize the multiple of 6: $\boxed{98,634}$.
98634
Level 2
Number Theory
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the greatest five-digit multiple of 6 using the digits 3, 4, 6, 8, and 9, we must first understand the conditions for a number to be a multiple of 6. A number is a multiple of 6 if it is divisible by both 2 and 3. 

For divisibility by 2, the last digit of the number must be even. The available even digits from our set are 4, 6, and 8. 

For divisibility by 3, the sum of the digits must be divisible by 3. The sum of the digits 3, 4, 6, 8, and 9 is 30. Since 30 is divisible by 3, any arrangement of these digits will satisfy the divisibility condition for 3.

Next, we need to maximize the five-digit number. To do this, we should start with the largest digits and place the smallest even digit at the end to ensure the number is even. 

The largest digit is 9, followed by 8, 6, 4, and 3. We will test the largest even digits for the last position. 

1. If we place 4 at the end, the number formed is 98634.
2. If we place 6 at the end, the number formed is 98436.
3. If we place 8 at the end, the number formed is 96438.

Among these combinations, 98634 is the largest. 

Thus, the greatest five-digit multiple of 6 that can be formed using the digits 3, 4, 6, 8, and 9 exactly once is 98634.<|END_OF_LATENT|> The final answer is $98634$.