Step-by-step Solution:

We are given a number \( N \) that, when divided by 9, 10, 15, and 20, leaves remainders of 4, 5, 10, and 4, respectively. This can be expressed as: \[ N \equiv 4 \pmod{9}, \quad N \equiv 5 \pmod{10}, \quad N \equiv 10 \pmod{15}, \quad N \equiv 4 \pmod{20}. \] To find the smallest possible \( N \), we first find the least common multiple (LCM) of the divisors 9, 10, 15, and 20. The LCM is calculated as: \[ \text{LCM}(9, 10, 15, 20) = \text{LCM}(9, 5, 4) = 180. \] Now, the number \( N \) can be expressed in the form: \[ N = 180n - 5. \] Substituting \( n = 2 \) gives: \[ N = 180 \times 2 - 5 = 360 - 5 = 355. \] Thus, the number is 355.