What values of k make x^2 + 12x + k factorable over the integers?

[judytl3]

try to determine all the positive values of k for which x^2 + 12x + k is factorable over the integers.

 

[Ackbach]

Here's my solution:

Spoiler

Set $x^{2}+12x+k=(x+a)(x+b)=x^{2}+(a+b)x+k.$
Evidently, then, the values of $k$ are products of numbers whose sum is $12$. The possibilities are as follows:
\begin{align*}
1+11=12& \to k=11\\
2+10=12& \to k=20\\
3+9=12& \to k=27\\
4+8=12& \to k=32\\
5+7=12& \to k=35\\
6+6=12& \to k=36.
\end{align*}
Then they repeat.