If integer x is chosen at random from 50 to 149 inclusive, what is the

\(x^3 - x\) can be written as \(x(x^2 - 1) = x (x + 1)(x - 1) = (x - 1) * x * (x + 1)\)

Basically, we want to know if the product of 3 consecutive terms is divisible by 12 for x = 50, 51...till 149

Every set of 3 consecutive numbers will always be divisible by 3. We need to check for divisibility by 4.

When x = 50, 49 * 50 * 51. Not divisible by 4

When x = 51, 50 * 51 * 52. Is divisible by 4

When x = 52, 51 * 52 * 53. Is divisible by 4

When x = 53, 52 * 53 * 54. Is divisible by 4

When x = 54, 53 * 54 * 55. Not divisible by 4

So we see that if x is even and not divisible by 4, then \(x^3 - x\) is not divisible by 12.

The series is 50, 54, 58.....146


The total numbers not divisible by 12 = \(\frac{146 - 50}{4} + 1 = 25\)

The total numbers = 149 - 50 + 1 = 100

The total numbers divisible by 12 = 100 - 25 = 75

Probability = Favorable outcomes / Total Outcomes = 75/100 = 3/4


Option D

Arun Kumar