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# For how many (not necessarily positive) integer values of n is the val

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Senior Manager
Status: Manager
Joined: 02 Nov 2018
Posts: 280
For how many (not necessarily positive) integer values of n is the val  [#permalink]

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18 Mar 2019, 06:04
1
00:00

Difficulty:

45% (medium)

Question Stats:

55% (02:17) correct 45% (02:40) wrong based on 11 sessions

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For how many (not necessarily positive) integer values of n is the value of $$4000.(\frac{2}{5})^n$$an integer?

(A) 3

(B) 4

(C) 6

(D) 8

(E) 9
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India
Re: For how many (not necessarily positive) integer values of n is the val  [#permalink]

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18 Mar 2019, 06:19
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For how many (not necessarily positive) integer values of n is the value of $$4000.(\frac{2}{5})^n$$an integer?

(A) 3

(B) 4

(C) 6

(D) 8

(E) 9

4000 = 2^5 * 5^3

So the denominator of the fraction $$(\frac{2}{5})^n$$ could be 1 or 5^1 or 5^2 or 5^3 or 2^1 or 2^2 or 2^3 or 2^4 or 2^5 for the expression to be an integer.

Hence n could be 0 (2/5 = 1/1)
n could be 1 [(2/5)^n = (2/5)]
n could be 2 [(2/5)^n = (2/5)^2]
n could be 3 [(2/5)^n = (2/5)^3]
n could be -1 [(2/5)^n = (2/5)^-1 = (5/2)^1]
and similarly, n could be -2, -3, -4 and -5

In all, we get 9 values of n.

_________________
Karishma
Veritas Prep GMAT Instructor

Re: For how many (not necessarily positive) integer values of n is the val   [#permalink] 18 Mar 2019, 06:19
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