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# If k is a positive integer, how many unique prime factors does 14k

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If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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Updated on: 11 May 2017, 23:31
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95% (hard)

Question Stats:

25% (01:47) correct 75% (01:41) wrong based on 66 sessions

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If k is a positive integer, how many unique prime factors does 14k have ?

(1) k^4 is divisible by 100
(2) 50*k has 2 prime factors

Give me an algebraic solution. And i will tell you where you went wrong

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Originally posted by stonecold on 22 Nov 2016, 10:24.
Last edited by hazelnut on 11 May 2017, 23:31, edited 3 times in total.
Renamed the topic and edited the question.
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 12:20
It depends on whether you're referring to prime factors or unique prime factors. I assume from the wording of statement 2 that it's the latter, but on the GMAT they'd have to be more specific.
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 12:23
ccooley wrote:
It depends on whether you're referring to prime factors or unique prime factors. I assume from the wording of statement 2 that it's the latter, but on the GMAT they'd have to be more specific.

Thank you.
I have edited the Post.
P.S => Its good to finally see someone from mgmat on this forum.
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 16:25
D

1) k has to be a multiple of 2 and 5. 14 carries the primes 2 and 7 so three primes

2. if 50 K only has two primes we can look what primes 50 can be broken down into
2 and 5 and 14 has 2 and 7 so again three primes
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 20:55
1
stonecold wrote:
if k is a positive Integer,how many unique Prime Factors Does 14k have ?

(1) k^4 is divisible by 100
(2)50*k has 2 Prime Factors

Give me an algebraic solution. And i will tell you where you went wrong

(1) $$k^4$$ is divisible by $$100=10^2$$, we have $$k^4=10^2 k'$$.

Since the left side is a double square of $$k$$ and the right side contains only a square of 10, we must have $$k'=10^2 \times k"^4$$.
So $$k^4=10^4 \times k"^4 \implies k=10k"$$ or $$k$$ is divisible by 10.

This means that $$k$$ has at least 2 prime factors 2 and 5, so $$14k=2 \times 7 \times k$$ has at least 3 prime factors 2, 5, and 7.

However, we cannot know the extract number of prime factors that $$14k$$ has. Insufficient.

(2) $$50k=2 \times 5^2 \times k$$. Since $$50=2 \times 5^2$$ has 2 prime factors 2 and 5 and $$50k$$ has only 2 prime factors, it is deduced that $$k$$ has at most 2 prime factors 2 and 5.

If $$k$$ has only one prime factors 2, $$14k$$ has 2 prime factors 2 and 7.
If $$k$$ has two one prime factors 2 and 5, $$14k$$ has 3 prime factors 2, 5 and 7.
Insufficient.

Combine (1) & (2):
(1): $$k$$ has at least 2 prime factors 2 and 5
(2): $$k$$ has at most 2 prime factors 2 and 5

(1)+(2): $$k$$ does have 2 prime factors 2 and 5.
So $$14k$$ has 3 prime factors 2, 5, and 7. Sufficient.

ccooley wrote:
It depends on whether you're referring to prime factors or unique prime factors. I assume from the wording of statement 2 that it's the latter, but on the GMAT they'd have to be more specific.

Please correct me, but I see no diffirence in these two sentences:
(1) 14k has 3 prime factors.
(2) 14k has 3 unique prime factors.
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 22:11
Bunnuel can you pls explain the answer

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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 22:39
nguyendinhtuong wrote:
stonecold wrote:
if k is a positive Integer,how many unique Prime Factors Does 14k have ?

(1) k^4 is divisible by 100
(2)50*k has 2 Prime Factors

Give me an algebraic solution. And i will tell you where you went wrong

(1) $$k^4$$ is divisible by $$100=10^2$$, we have $$k^4=10^2 k'$$.

Since the left side is a double square of $$k$$ and the right side contains only a square of 10, we must have $$k'=10^2 \times k"^4$$.
So $$k^4=10^4 \times k"^4 \implies k=10k"$$ or $$k$$ is divisible by 10.

This means that $$k$$ has at least 2 prime factors 2 and 5, so $$14k=2 \times 7 \times k$$ has at least 3 prime factors 2, 5, and 7.

However, we cannot know the extract number of prime factors that $$14k$$ has. Insufficient.

(2) $$50k=2 \times 5^2 \times k$$. Since $$50=2 \times 5^2$$ has 2 prime factors 2 and 5 and $$50k$$ has only 2 prime factors, it is deduced that $$k$$ has at most 2 prime factors 2 and 5.

If $$k$$ has only one prime factors 2, $$14k$$ has 2 prime factors 2 and 7.
If $$k$$ has two one prime factors 2 and 5, $$14k$$ has 3 prime factors 2, 5 and 7.
Insufficient.

Combine (1) & (2):
(1): $$k$$ has at least 2 prime factors 2 and 5
(2): $$k$$ has at most 2 prime factors 2 and 5

(1)+(2): $$k$$ does have 2 prime factors 2 and 5.
So $$14k$$ has 3 prime factors 2, 5, and 7. Sufficient.

ccooley wrote:
It depends on whether you're referring to prime factors or unique prime factors. I assume from the wording of statement 2 that it's the latter, but on the GMAT they'd have to be more specific.

Please correct me, but I see no diffirence in these two sentences:
(1) 14k has 3 prime factors.
(2) 14k has 3 unique prime factors.

Hi
I think i can answer that.
In mgmat books the explanation is given as follows =>
n=2*3^2 => two unique prime factors and 3 Total prime factors.
They refer to total Prime factors as "length" where we just have to add the exponents.
It just Creates confusion if you ask me since in GMAT/general mathematics we always write prime factors as to unique prime factors.

Regards
Stone Cold
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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22 Nov 2016, 22:51
1
stonecold wrote:
nguyendinhtuong wrote:
ccooley wrote:
It depends on whether you're referring to prime factors or unique prime factors. I assume from the wording of statement 2 that it's the latter, but on the GMAT they'd have to be more specific.

Please correct me, but I see no diffirence in these two sentences:
(1) 14k has 3 prime factors.
(2) 14k has 3 unique prime factors.

Hi
I think i can answer that.
In mgmat books the explanation is given as follows =>
n=2*3^2 => two unique prime factors and 3 Total prime factors.
They refer to total Prime factors as "length" where we just have to add the exponent.
It just Creates confusion if you ask me since in GMAT/general mathematics we always wrote prime factors as to unique prime factors.

Regards
Stone Cold

Still confused As I see, $$n=2*3^2$$ still has only 2 prime factors 2 and 3.

Oh, I got this. In MGMAT, they stated that: 72 splits into 5 total prime factors: 2x3x2x2x3. However, 72 still does only 2 prime factors 2 and 3.

Hence, your question meets no confusion between "prime factors" and "unique prime factors".
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Re: If k is a positive integer, how many unique prime factors does 14k  [#permalink]

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16 Dec 2017, 11:49
stonecold wrote:
If k is a positive integer, how many unique prime factors does 14k have ?

(1) k^4 is divisible by 100
(2) 50*k has 2 prime factors

Give me an algebraic solution. And i will tell you where you went wrong

Bunuel can you help us out? lol
Re: If k is a positive integer, how many unique prime factors does 14k   [#permalink] 16 Dec 2017, 11:49
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