anonymous
  • anonymous
For how many positive integers for k is the value of k^2 - 18k + 65 a positive prime number? 0, 1, 2, or 4?
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
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asnaseer
  • asnaseer
see if you can factorise the expression:\[k^2-18k+65\]if you can then this implies it can only generate a prime number if one of the factors is equal to 1.
anonymous
  • anonymous
it factors into (x-5)(x-13) and how do you know that it can only generate a prime number if its 1? Oh! Any integer squared is automatically composite unless its 1. So we try 1 and -1 and we get 48 and 84 respectively. So, the answer is 0? But even if something is squared and it is composite, it could still be changed into prime by the -18 or the + 65
anonymous
  • anonymous
Well, not by the -18, but it could indirectly help the 65 make f(k) prime.

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phi
  • phi
In general x*y is composite (by definition) unless x=1 and y is prime or y=1 and x is prime. for your case (with integer k>0) (k-5)(k-13) we ask can (k-5) = 1 and (k-13) be prime and vice versa for k=4, we get -1* -9 = 9 not prime k=6 1* -7 negative but the question asks for positive primes k= 12 7 * -1 neg k=14 9 * 1 not prime
phi
  • phi
It looks like zero primes
anonymous
  • anonymous
How come we want to make the x-intercepts = 1? Doesnt that leave out a whole bunch of other numbers that we could try into the original equation?
anonymous
  • anonymous
the answer is 0 but i dont understand how you arrived at that
phi
  • phi
Do you agree with this statement: x*y is composite (by definition) unless x=1 and y is prime or y=1 and x is prime.
phi
  • phi
k^2 - 18 k + 65 = (k-5)(k-13)
anonymous
  • anonymous
Yes, because after the multiplication, that product would be divisible by both x and y
phi
  • phi
so x=(k-5) and y= (k-13)
anonymous
  • anonymous
?
phi
  • phi
x*y is composite unless.... (k-5)*(k-13) is composite unless....
anonymous
  • anonymous
Unless one of those equals 1! Oh. I see now. Goodness I can be a real stonehead sometimes heheh. Thanks
phi
  • phi
I was getting nervous, cause I do not know how to say it any other way

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