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If k is a positive integer whose only prime factors are 2 and 5, is k

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Bunuel
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If k is a positive integer whose only prime factors are 2 and 5, is k [#permalink] New post   27 Nov 2022, 12:15
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A
B
C
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E

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74% (01:39) correct 26% (01:52) wrong based on 39 sessions
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If k is a positive integer whose only prime factors are 2 and 5, is k divisible by 20?

(1) k is a multiple of 16
(2) k has 6 factors
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A
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Re: If k is a positive integer whose only prime factors are 2 and 5, is k [#permalink] New post   27 Nov 2022, 13:15
Bunuel wrote:
If k is a positive integer whose only prime factors are 2 and 5, is k divisible by 20?

(1) k is a multiple of 16
(2) k has 6 factors


Given

As 2 and 5 are prime factors of k, we can write

k = 10*x

x is a quotient when k is divided by 10.

Statement 1

(1) k is a multiple of 16

We know k is a multiple of 5. As k is a multiple of 16, it is a multiple of 4.

As k is a multiple of both 4 and 5, k is a multiple of 20.

The statement is sufficient. We can eliminate B, C and E.

Statement 2

(2) k has 6 factors

k can be \(2^2 * 5^1\) | Is k divisible by 20 ? Yes

k can be \(2^1 * 5^2\) | Is k divisible by 20 ? No

As there are two different answers, we can eliminate B.

Option A
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Re: If k is a positive integer whose only prime factors are 2 and 5, is k [#permalink] New post   28 Nov 2022, 02:03
k has prime factors 2 & 5
target is k divisible by 20 ; 2^2*5
#1
k is a multiple of 16
k = 2 *5* x where x has to be least 2^3 or its multiple
2*5*x / 2^2 * 5
x would be 8 or its multiple ; sufficient to say that k is divisible by 20
#2
k has 6 factors
2^a * 5^b

case 1 ; a is 2 then b = 1 ; we get yes to target
case 2 : a is 1 and b = 2 ; we get no to target

insufficient
option A

Bunuel wrote:
If k is a positive integer whose only prime factors are 2 and 5, is k divisible by 20?

(1) k is a multiple of 16
(2) k has 6 factors
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Bunuel
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Re: If k is a positive integer whose only prime factors are 2 and 5, is k [#permalink] New post   09 Jan 2023, 00:00
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Bunuel wrote:
If k is a positive integer whose only prime factors are 2 and 5, is k divisible by 20?

(1) k is a multiple of 16
(2) k has 6 factors


If k is a positive integer whose only prime factors are 2 and 5, is k divisible by 20?

k is a positive integer whose only prime factors are 2 and 5: \(k=2^x*5^y\), where \(x\geq{1}\) and \(y\geq{1}\) (notice that the number of factors of k would be (x + 1)(y + 1)). Also notice that k will be divisible by \(20 = 2^2*5\) if \(x\geq{2}\) and \(y\geq{1}\) (this condition is already satoisfied from the stem). So, the question basically asks whether \(x\geq{2}\).

(1) k is a multiple of 16. 16 = 2^4, hence this statement implies that \(x \geq{4}\). Sufficient.

(2) k has 6 factors --> this means that (x + 1)(y + 1) = 6. If x = 2 and y = 1, then the answer will be YES but if x = 1 and y = 2, then the answer will be NO. Not sufficient.

Answer: A.

Hope it's clear.
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Re: If k is a positive integer whose only prime factors are 2 and 5, is k [#permalink]
09 Jan 2023, 00:00
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Bunuel
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