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

Sagot :

Konrad509idn
[tex]n^2+6n-27=\\ n^2-3n+9n-27=\\ n(n-3)+9(n-3)=\\ (n+9)(n-3)[/tex]
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.

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

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

So, [tex]n^2+6n-27[/tex] is a prime number for [tex]n=4[/tex]


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