What is the largest five-digit integer whose digits have a product equal to the product (7) (6)(5)(4)(3)(2)(1)?

Just re-arrange your 7 numbers slightly by multiplying 2 x 4 = 8 and put them in order from the biggest to the smallest and you get this: 87,653.

We want the biggest numbers, and we can get a 9 from $6\times 3$ which is $9\times 2$ .

Therefore, now we have the biggest digit as 9 with 2, 2, 4, 5, and 7 left over.

The next largest digit is probably 8 as $2\times 4=8$ , leaving us with 2, 5, and 7.

So far we have 98___ and we can see that $2\times 5=10, 2\times 7=14, 5\times7=35$and the tens digits are all less than 7 so now we go decreasing order and get

98752.

You are very welcome!

:P