When you divide me by 5 and get a remainder 4. I have two digits. both odd. the sum of my digits 10

From the first line we know that the correct answer is 4 more than a multiple of 5.

Since we also know it is a two digit number, and all multiples of 5 in base ten end with a 5 or a 0, we know the second digit must be either 4 or 9. However, since it must be odd, 9 is the only option. Finally, subtract that from ten to get the other digit (1, which is odd so we are good) and the answer is 19