### Instruction: What is the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$?### Response: I need to find a five-digit integer that has the same product of digits as 7 factorial, which is 5040.
I want the digits to be as large as possible, so I will start with 9 and see how many times it divides 5040.
5040 divided by 9 is 560, so I can use 9 as the first digit.
Now I have 560 left to distribute among the other four digits.
I will try to use 8 as the next digit, since it is the next largest after 9.
560 divided by 8 is 70, so I can use 8 as the second digit.
Now I have 70 left to distribute among the last three digits.
I will try to use 7 as the next digit, since it is the next largest after 8.
70 divided by 7 is 10, so I can use 7 as the third digit.
Now I have 10 left to distribute among the last two digits.
I will try to use 5 as the next digit, since it is the next largest after 7.
10 divided by 5 is 2, so I can use 5 as the fourth digit.
Now I have 2 left for the last digit, which is easy.
The last digit is 2.
Therefore, the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$ is 98752.