Form two four-digit numbers from the digits 2, 3, 4, 5, 6, 7, 8,
and 9, using each digit exactly once, so that the absolute value of
the difference of the two numbers is as small as possible.
To find two four-digit numbers with the smallest absolute difference, we can start by arranging the digits in descending order to create a larger number and in ascending order to create a smaller number.
Step 1: Sort the digits in descending order:
9, 8, 7, 6, 5, 4, 3, 2
Step 2: Form the larger number:
Starting from the left, take the digits one by one:
Find the largest possible digit for the thousands place, which is 9.
Next, find the largest possible digit for the hundreds place, which is 8.
Continue this process until you have formed a four-digit number.
The larger number formed is 9876.
Step 3: Form the smaller number:
Sort the digits in ascending order:
2, 3, 4, 5, 6, 7, 8, 9
Using the same process as in Step 2, form a four-digit number.
The smaller number formed is 2345.
Step 4: Find the absolute difference:
Take the absolute difference between the larger number (9876) and the smaller number (2345).
|9876 - 2345| = 7531
Therefore, the two four-digit numbers with the smallest absolute difference are 9876 and 2345, and the absolute difference is 7531.