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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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New post 29 Oct 2017, 15:26
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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
[Reveal] Spoiler: OA

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Kudos [?]: 65 [0], given: 42

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

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New post 29 Oct 2017, 17:07
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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.

Answer B

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

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New post 29 Oct 2017, 17:38
\(\frac{26k}{84}\) 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)

Answer: B.
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Re: If K is a positive integer less than 205, and...   [#permalink] 29 Oct 2017, 17:38
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