Junior Mathematical Olympiad 2025 Section A Q6

The sum of the factors of $2025$ is $1 + 3 + 5 + 9 + 15 + 25 + 27 + 45 + 75 + 81 + 135 + 225 + 405 + 675 + 2025 = 3751,$ which is an odd number. What is the smallest number $N > 2025$ for which the sum of the factors of $N$ is odd?

This is an Olympiad problem, so you must show that your value of $N$ has an odd sum of factors, and that no other value between $2025$ and $N$ does.