Minimum Possible Value of Natural Number n

Let the three integers be $a, b, c$. Sum of the three integers $= a + b + c = 3 \times 13 = 39$.

When $n$ is added, the new sum is $39 + n$, and the average of these four numbers is: $\text{Average} = \frac{39 + n}{4}$

We are given that this average is an odd integer, say $2k + 1$ where $k$ is an integer. $\frac{39 + n}{4} = 2k + 1$ $39 + n = 8k + 4$ $n = 8k - 35$

Since $n$ is a natural number ($n \ge 1$): $8k - 35 \ge 1 \implies 8k \ge 36 \implies k \ge 4.5$

Since $k$ must be an integer, the minimum value of $k$ is $5$.

Then $n = 8(5) - 35 = 40 - 35 = 5$.

Check: Average $= \frac{39 + 5}{4} = 11$, which is an odd integer. Thus, the minimum possible value of $n$ is $5$.