Arun selected an integer $$x$$ between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer $$x$$ and any other integer between 2 and 40, both inclusive, is 1.
How many different choices for such an $$x$$ are possible?

We know that if the greatest common divisor of two numbers is 1, they are co-prime to each other.

Given that the greatest common divisor of the selected integer $$x$$ and any other integer between 2 and 40, both inclusive, is 1.

So, the selected number has to be a prime number, and that too greater than $$\dfrac{40}{2}$$ or $$20$$ because prime numbers less than 20, like 17, will have the greatest common divisor = 17 with their multiple, like 34 which is also in the range of selected integers.

Hence, the only possible values of $$x$$ are 23, 29, 31 and 37 i.e. a total of 4 values.