### Instruction: Each of the digits 3, 4, 6, 7, 8, and 9 is used once and only once to form a six-digit integer. What is the largest six-digit multiple of 4 that can be formed?### Response: To form the largest six-digit multiple of 4, I want to maximize the value of each digit, starting from the leftmost one.
The leftmost digit has the largest place value, so I want to use the largest available digit there, which is 9.
The next digit has the second largest place value, so I want to use the second largest available digit there, which is 8.
The next digit has the third largest place value, so I want to use the third largest available digit there, which is 7.
Now I have used 9, 8, and 7, and I have 3, 4, and 6 left.
To form a multiple of 4, the last two digits must form a multiple of 4.
The largest multiple of 4 that I can form with 3, 4, and 6 is 64, so I want to use those as the last two digits.
That leaves me with 3 as the fourth digit.
Therefore, the largest six-digit multiple of 4 that I can form is 987,364.