GeminiHeat wrote:
x is divisible by 144. If \(\sqrt[3]{x}\) is an integer, then which of the following is \(\sqrt[3]{x}\) definitely divisible by?
I. 4
II. 8
III. 12
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
If \(x\) is divisible by \(144\) then, \(x\) must have all the factors of it
\(144 = 12^2 = 4^23^2 = 2^43^2\)
So \(x = 2^43^2(k)\)
Cube-rooting both the sides:
\(\sqrt[3]{x} = \sqrt[3]{2^43^2(k)}\)
\(\sqrt[3]{x} = 2\sqrt[3]{2^13^2(k)}\)
This means \(k\) must have two \(2\)s and one \(3\)
\(k = 12\)
So \(x = 2^43^2(12) = 12^3\)
Therefore, \(\sqrt[3]{x} = 12\) which will always be divisible by \(4\) and \(12\)
Hence, option E