Is it or isn't it?

If $n$ is expressable as a sum of $3$ consecutive integers, then $n=k_1 + (k_1 + 1) + (k_1 + 2) = 3k_1 + 3 = 3(k_1 + 1)$ We see that $n$ is divisible by $3$ .

Now, if $n$ is expressable as a sum of $7$ consecutive integers, then $n= k_2 + (k_2 + 1)+ (k_2 + 2)+ (k_2 + 3)+ (k_2 + 4)+ (k_2 + 5)+ (k_2 + 6)=7k_2 + 21 = 7(k_2 + 3)$ We see that $n$ is divisible by $7$ .

Finally, if $n$ is expressable as a sum of $10$ consecutive integers, then $\begin{aligned} n &= k_3 + (k_3 + 1) + (k_3 + 2) + (k_3 + 3) + (k_3 + 4) + (k_3 + 5) + (k_3 + 6) + (k_3 + 7) + (k_3 + 8) + (k_3 + 9) \\ &= 10k_3 + 45 = 5(2k_3 + 9) \end{aligned}$ We see that $n$ is divisible by $5$ . Furthermore, we note from here that $n$ must be odd (since $2k_3 + 9$ is always odd for all integers $k_3$ and $5$ is odd).

To summarize, $n$ should be divisible by $3,7,5$ and $n$ should be odd. The smallest such number is $n=3\times 5\times 7 = 105$ . However, assuming $n$ is expressable as a sum of $9$ consecutive odd integers, $\begin{aligned} n &= k_4 + (k_4 + 2) + (k_4 + 4)+(k_4 + 6)+(k_4 + 8)+(k_4 + 10)+(k_4 + 12)+(k_4 + 14)+(k_4 + 16)\\&=9k_4 + 72 = 9(k_4 + 8) \end{aligned}$ then $n$ must be divisible by $9$ . But $105$ is not divisible by $9$ , a contradiction.

Thus, $n$ is not expressable as a sum of $9$ consecutive odd integers.