Use all the digit 5, 8, 1, 2, and 7 once to form the greatest 5-digit odd number.

To make the greatest 5-digit number, we need to put the greatest digit on the most left. We have 8, 7, 5, 2, and 1. So, the greatest 5-digit number is 87521. Coincidentally, 1 is an odd number. So, the greatest 5-digit odd number possible is 87521.