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

Question

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

gmatophobia · Answer

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

Archit3110 · Answer

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 · Answer

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.

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

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

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