Question

Find all integers n for which n^2+6n-27 is a prime number? Please tell me how you did it.

Answer 1

$n^2+6n-27=\\ n^2-3n+9n-27=\\ n(n-3)+9(n-3)=\\ (n+9)(n-3)$
For the above product to be a prime number, one of the factors must be a prime number and the other must be equal to 1.

$n+9=1\\ n=-8\\\\ -8-3=-11$
The first factor is equal 1 for $n=-8$, but the other is euqal -11, which is not a prime number.

$n-3=1\\ n=4\\\\ 4+9=13$
The second factor is equal 1 for $n=4$ and the first factor is equal 13, which is a prime number.

So, $n^2+6n-27$ is a prime number for $n=4$