How many 5-digit positive integers exist the sum of whose digits are odd?

There are 9 × 10⁴ = 90000, 5-digit positive integers.

Out of these 90000 positive integers, the sum of the digits of half of the numbers will add up to an odd number and the remaining half will add up to an even number.

Hence, there are $$\frac{{90000}}{2}$$  = 45000, 5-digit positive integers whose sum add up to an odd number.