Final answer:
The greatest multiple of 36 that uses all ten digits exactly once is 9876543012. This number is divisible by both 4 and 9, fulfilling the criteria for divisibility by 36.
Step-by-step explanation:
The greatest multiple of 36 that uses all ten digits exactly once can be found through a process of trial and error, ensuring that the resulting number is divisible by 36. To be divisible by 36, a number must be divisible by both 4 and 9. The divisibility rule for 4 is that the last two digits of the number must form a number divisible by 4. The divisibility rule for 9 is that the sum of all the digits must be divisible by 9.
Based on these rules, the largest such number would start with the largest digits and end with two digits that form a multiple of 4. After many trials, we deduce that the greatest multiple of 36 using each digit 0-9 exactly once is 9876543012. This number uses all ten digits once, the sum of its digits is 45 (which is divisible by 9), and the last two digits (12) form a number that is divisible by 4. Therefore, 9876543012 is indeed a multiple of 36.