SOLUTION: How many positive integers n are there such that nē - 14n + 24 is a prime number?

Algebra.Com
Question 550071: How many positive integers n are there such that nē - 14n + 24 is a prime number?
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
A prime number is an integer greater than 1 
that has no other factors beside itself and 1

nē - 14n + 24

can be factored as

(n - 2)(n - 12)

Therefore unless one of the factors (n - 12) or (n - 2) is 1, and the other
a prime number, (n - 2)(n - 12) will not be prime.

If (n - 2) equals 1 then

    n - 2 = 1
        n = 3

But (3 - 2)(3 - 12) = 1(-9) = -9

But that is a negative number so (n - 2) can't be the factor that is 1.

If (n - 12) equals 1 then

    n - 12 = 1
        n = 13

And (13 - 2)(13 - 12) = 11(1) = 11 which IS prime.

So 13 is the only number n can be such 
that nē - 14n + 24 is prime.

The questios is how many positive integers is that? 
The answer is one, the number 13.

Edwin
RELATED QUESTIONS

How many positive integers n are such that the value of the expression n^3 -14n^2 + 64n... (answered by richard1234)
there are some positive integers n such that n^3 -7 is divisible by n-3 . how many such... (answered by drk)
How many positive integers n are there such that 2n+1 is a divisor of... (answered by ikleyn,greenestamps)
if n>1 and each of the three integers n, n+2,and n+4 is a prime number, then the set of... (answered by stanbon)
How many positive integers n < 2022 are there for which the sum of the odd positive... (answered by greenestamps)
For arbitrary positive integers k, m, and n, will there exist a prime p such that... (answered by Edwin McCravy)
For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than... (answered by ikleyn,math_tutor2020)
Determine all integers n such that n^4-4n^3+15n^2-30n+27 is a prime... (answered by KMST)
Let $P(x)$ be a nonconstant polynomial, where all the coefficients are nonnegative... (answered by jim_thompson5910)