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# If K is a positive integer less than 205, and...

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If K is a positive integer less than 205, and... [#permalink]

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29 Oct 2017, 16:26
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25% (medium)

Question Stats:

70% (01:13) correct 30% (00:43) wrong based on 27 sessions

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If $$k$$ is a positive integer less than 205, 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

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[Reveal] Spoiler: OA

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If K is a positive integer less than 205, and... [#permalink]

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29 Oct 2017, 18:07
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sasyaharry wrote:
If $$k$$ is a positive integer less than 205, 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

Prime factorize 26 = $$2^113^1$$
Prime factorize 84 = $$2^23^17^1$$

We have
$$\frac{26k}{84}=\frac{(2*13)*k}{2^2*3*7}=$$ an integer

For the quotient to be an integer, every power of every prime factor of 84 must be included in the numerator.

The numerator is missing:
one 2 (there is one in 26), one 7, and one 3. Those are the factors of $$k$$. So

$$\frac{(2*13) *(2*3*7)}{2^2*3*7}=\frac{(26)(42)}{84}=\frac{1092}{84} = 13$$

$$k = 42 = 2*3*7$$. Three different primes.

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Re: If K is a positive integer less than 205, and... [#permalink]

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29 Oct 2017, 18:38
$$\frac{26k}{84}$$ is an integer

K must be multiple of 84
84*2= 168

Prime factors of 168 are: $$2^3$$*3*7

K has 3 different positive prime factors (2, 3 and 7)

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Re: If K is a positive integer less than 205, and...   [#permalink] 29 Oct 2017, 18:38
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