what is the largest multiple of 12 that can be written using each digit 0,1,2,3,4,5,6,7,8,9 exactly once?
9 876 543 120
To find the largest multiple of 12 that can be written using each digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once, we need to understand the divisibility rules of 12 and find a suitable arrangement of the digits.
Divisibility rule for 12: A number is divisible by 12 if it is divisible by both 3 and 4.
Step 1: Arrange the digits in descending order: 9876543210
Step 2: Find the sum of the digits: 9+8+7+6+5+4+3+2+1+0 = 45
Step 3: Check if the sum of the digits is divisible by 3. In this case, 45 is divisible by 3, so the arrangement of digits is divisible by 3.
Step 4: Starting from the rightmost digit, check divisibility by 4. The last two digits, 10, are not divisible by 4. So we move left one digit and check the last two digits again (21). This is also not divisible by 4. We continue this process until we find a pair of two digits that is divisible by 4.
Step 5: The pair of two digits before the last digit is 43, which is divisible by 4. Hence, our answer is 9876543120.
Therefore, the largest multiple of 12 that can be written using each digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once is 9876543120.