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Integers

by Akansha » Thu May 26, 2011 9:45 am
Positive integer k has exactly 2 positive prime factors, 3 and 7. If k has a
total of 6 positive factors, including 1 and k, what is the value of k?
a. 3^2 is a factor of k
b. 7^2 is not a factor of k

OA is D
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by clock60 » Thu May 26, 2011 10:54 am
here k can be performed two ways
k=(3^2)*7^1. number of factors (2+1)*(1+1)=6 or
k=(7^2)*3, with the same number of factors
(1) says that 3^2 is a factor of k. only k=(3^2)*7^1 possible suff
(2)says that k is has no factor 7^2, thus it can`t be k=7^2*3^1, and so it must be (3^2)*(7^1) also suff
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by Anurag@Gurome » Thu May 26, 2011 7:32 pm
Akansha wrote:Positive integer k has exactly 2 positive prime factors, 3 and 7. If k has a
total of 6 positive factors, including 1 and k, what is the value of k?
a. 3^2 is a factor of k
b. 7^2 is not a factor of k

OA is D

Solution:
The integer k can be represented as (3^a)*(7^b) where a and b are certain positive integers.
Now, total number of factors of k is (a+1)*(b+1) = 6.
So the possible combinations of (a+1) and (b+1) are (1, 6),(2,3),(3,2), (6,1).
Or the possible combinations of a and b are (0,5), (1,2), (2, 1), (5,0).
Now, (0,5) and (5,0) are eliminated because both a and b have to be positive.
Let us now consider statement a. alone.
It means that a >= 2. Or the only feasible option is a = 2 and b = 1.
So, k = (3^2)*7 = 63.
Next consider statement b. alone.
It means that b < 2. Again the only feasible option is a = 2 and b = 1.
Or k = 63.
So, each statement alone is sufficient to answer the question.

The correct answer is (D).
Anurag Mairal, Ph.D., MBA
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by Ozlemg » Fri May 27, 2011 8:42 am
Thank you very much for your explanation.
But I did not understand why "total number of factors of k is (a+1)*(b+1) equals to 6"?
Why dowe add +1 to a and b?

Thank you.
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