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If k is a positive integer less than 215

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If k is a positive integer less than 215  [#permalink]

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New post Updated on: 10 Mar 2018, 06:15
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Question Stats:

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If k is a positive integer less than 215, and \(\frac{26k}{84}\) is an integer, how many different positive prime factors k can have?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

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Originally posted by itisSheldon on 10 Mar 2018, 02:00.
Last edited by itisSheldon on 10 Mar 2018, 06:15, edited 1 time in total.
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Re: If k is a positive integer less than 215  [#permalink]

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New post Updated on: 10 Mar 2018, 02:48
84 = 2*2*3*7
26=13*2
to be the term integer ,K can be 2*3*7*5

Originally posted by sobby on 10 Mar 2018, 02:23.
Last edited by sobby on 10 Mar 2018, 02:48, edited 1 time in total.
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Re: If k is a positive integer less than 215  [#permalink]

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New post 10 Mar 2018, 02:41
Hi sobby

K could be 2*3*5*7 = 210(which is less than 215)
The rest of your solution remains the same.
84 can be prime factorized as 2*2*3*7 and 26 as 2*13. 26K/84 can be an integer even if K is 210.

Therefore, K can have a total of 4 prime factors(Option C)
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Re: If k is a positive integer less than 215  [#permalink]

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New post 10 Mar 2018, 02:47
pushpitkc wrote:
Hi sobby

K could be 2*3*5*7 = 210(which is less than 215)
The rest of your solution remains the same.
84 can be prime factorized as 2*2*3*7 and 26 as 2*13. 26K/84 can be an integer even if K is 210.

Therefore, K can have a total of 4 prime factors(Option C)


ahh , multiplication error ... but still i think , question must say maximum number different of prime factor...
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Re: If k is a positive integer less than 215  [#permalink]

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New post 10 Mar 2018, 03:02
sobby wrote:
pushpitkc wrote:
Hi sobby

K could be 2*3*5*7 = 210(which is less than 215)
The rest of your solution remains the same.
84 can be prime factorized as 2*2*3*7 and 26 as 2*13. 26K/84 can be an integer even if K is 210.

Therefore, K can have a total of 4 prime factors(Option C)


ahh , multiplication error ... but still i think , question must say maximum number different of prime factor...


Yeah, I agree. Properly wording the question would remove the ambiguity.
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Re: If k is a positive integer less than 215  [#permalink]

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New post 10 Mar 2018, 03:58
itisSheldon wrote:
If k is a positive integer less than 215, and \(\frac{26k}{84}\) is an integer, how many different positive prime factors does k have?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6


IMO B or C. I went ahead with B and got it wrong.

26k /84 is an integer.... hence 42 divides k. 42 = 2*3*7. SO K can has a min of 3 factors... here however we can introduce another prime factor ( unique ofcourse ) of 5 and get 42 * 5 = 210.

So I think the question is very ambiguous either K should be less than a number N such that we cannot include 5 as a prime factor or question should be re-phrased to end with "...how many different prime factors can k have?"

Please give kudos if you liked my explanation.
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If k is a positive integer less than 215  [#permalink]

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New post 10 Mar 2018, 11:25
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itisSheldon wrote:
If k is a positive integer less than 215, and \(\frac{26k}{84}\) is an integer, how many different positive prime factors k can have?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6


let x=26k/84
k=84x/26→42x/13
42x/13<215
x<66.5
try 66 no
try 65 yes
(42*65)/13=210=k
prime factors of 210 are 2,3,5,7
4
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If k is a positive integer less than 215   [#permalink] 10 Mar 2018, 11:25
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