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If K is a positive integer less than 205, and...
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29 Oct 2017, 16:26
29 Oct 2017, 16:26
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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
A) 2
B) 3
C) 4
D) 5
E) 6
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Re: If K is a positive integer less than 205, and...
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29 Oct 2017, 18:07
29 Oct 2017, 18:07
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Expert Reply
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
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
Re: If K is a positive integer less than 205, and...
[#permalink]
29 Oct 2017, 18:38
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)
Answer: B.
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.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
