+53 What is the product of the two integer values for x for which |x^2 - 16| is a prime number?
The two numbers can be +3 and -5. Also -9 and -11 will work. There maybe an infinite number of them.
abs[3^2 - 16] = 7 which is a prime number.
abs[-5^2 - 16] =41 which is a prime number.
+3 x -5 =-15
No, there is only a finite number of those numbers.
proof: suppose |x2-16| is a prime number. |x2-16|=|x-4|*|x+4|, so one of the multiplied numbers=1 (because the product is a prime number).
|x-4|=1 means x=5 or x=3, and |x+4|=1 means x=-5 or x=-3. if x=5 then |x+4|=9, and because 9 is not a prime number (9=3*3) x cannot be equal to 5.
if x=-5 then |x-4|=9, so x cannot be equal to -5.
so if |x2-16| is a prime number, then x=3 or x=-3. the product of the solutions is 3*-3=-9, and that's the answer.
Do you mean: abs[-5^2 - 16] =9? If so, why is W/A giving 41 as the answer? This is how they come to that answer:
Simplify the following:
abs(-5^2 - 16)
Evaluate 5^2.
5^2 = 25:
abs(-25 - 16)
Evaluate -25 - 16.
-25 - 16 = -41:
abs(-41)
Absolute value gives the distance to the origin.
Since -41<=0, then abs(-41) = 41:
=41
https://www.wolframalpha.com/input/?i=abs%5B-5%5E2+-+16%5D+%3D%3F
