Bunuel wrote:
x is a positive integer less than 30, which of the following option correctly indicates the number of values of x for which x^2/18 has 2 factors?
A. 0
B. 1
C. 4
D. 5
E. 7
Are You Up For the Challenge: 700 Level QuestionsAt first I approached this question by tabulating all the scenarios, but it took quite a lot of time. So, I decided to solve this question purely with inferences to see if it would save me some time. And yes it did!
So, here is my approach (not sure if I missed anything in this series of inferences
, pls let me know if you find gaps
)
Constraints: x: integer; 1<=x <30
say A =x^2/18 has 2 factors means A = prime number (property/definition of Prime)
1. For A to be a Prime: A must be an integer first. So \(x^2\) must be divisible by 18 = 3^2 * 2. Thus, x must be divisible by 2.
2. Say x = 2k --> \(x^2 \)= \((2k)^2 \)= \(4*k^2\); A = \(\frac{4k^2}{3^2 *2}\) = \(\frac{2 * k^2}{3^2}\)
3. Because A must be an integer, k^2 must be divisible by 3^2; thus \(k^2\) = \(3^2 \)* n
then, we have A = \(\frac{2 * 3^2 * n}{3^2}\) <=> A=2n always even
4. Meanwhile A is a prime and there is only 1 even prime which is 2 --> thus A=2 and there is only 1 value of x resulting in A=2
(there could be 2 values of x= 6 or -6 to have A=2; but don't forget we have a constraint that x is positive integer)Hence, answer is B: 1 value.
Hope this helps.