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If \(x\) is a number such that \(0 < x \le 20,\) for how many values of \(x\) is \(\dfrac{20}{x}\) an integer?

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If \(x\) is a number such that \(0 < x \le 20,\) for how many values of \(x\) is \(\dfrac{20}{x}\) an integer?

by Gmat_mission » Wed Jun 24, 2020 7:11 am

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If \(x\) is a number such that \(0 < x \le 20,\) for how many values of \(x\) is \(\dfrac{20}{x}\) an integer?

(A) Four
(B) Six
(C) Eight
(D) Ten
(E) More than ten

[spoiler]OA=E[/spoiler]

Source: Manhattan GMAT
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Re: If \(x\) is a number such that \(0 < x \le 20,\) for how many values of \(x\) is \(\dfrac{20}{x}\) an integer?

by Scott@TargetTestPrep » Mon Jun 29, 2020 8:40 am
Gmat_mission wrote: ↑
Wed Jun 24, 2020 7:11 am
 If \(x\) is a number such that \(0 < x \le 20,\) for how many values of \(x\) is \(\dfrac{20}{x}\) an integer?

(A) Four
(B) Six
(C) Eight
(D) Ten
(E) More than ten

[spoiler]OA=E[/spoiler]

Source: Manhattan GMAT
Solution:

If x is an integer, then 20/x is also an integer if x is a factor of 20. However, since it’s not mentioned that x has to be an integer, we see that if x = 1/n where n is a positive integer, we have 20/x = 20/(1/n) = 20n will always be an integer. Since there are infinitely many values of n, there are infinitely many values of x such that 20/x is an integer.

Answer: E

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