Rule 1:Stuti567 wrote:What is the number of positive integers that divide k^2 but do not divide k, where k is a positive integer?
(1) k^2 has a total of 13 factors
(2) √k has a total of 4 factors
For any positive integer k such that k = (x^a)(y^b)(z^c) -- where x, y and z are distinct prime numbers and a, b and c are positive integers -- then the number of positive factors of k = (a+1)(b+1)(c+1).
Follow the colors:
If k = 2³, then the number of positive factors of k = 3+1 = 4.
If k = 2³3², then the number of positive factors of k = (3+1)(2+1) = 12.
If k = 2³3²5�, then the number of positive factors of k = (3+1)(2+1)(7+1) = 96.
For more on this rule, check my post here:
https://www.beatthegmat.com/all-factors- ... 15019.html
Rule 2:
Since k is a factor of k², any factor of k must also be a factor of k².
Statement 1: k² has a total of 13 factors
Since 13 is a prime number, any integer with a total of 13 factors must be in the following form:
x¹², with the result the number of factors = 12+1 = 13.
Thus:
k² = x¹², with the result that the number of factors = 12+1 = 13.
k = x�, with the result that the number of factors = 6+1 = 7.
According to Rule 2, the 7 factors that divide k must also divide k².
Thus:
Total number of positive integers that divide k² but not k = (total number of factors of k²) - (total number of factors of k) = 13-7 = 6.
SUFFICIENT.
Statement 2: √k has a total of 4 factors
Case 1: √k = x³, with the result that the number of factors = 3+1 = 4.
Since √k = x³, we get:
k = x�, with the result that the number of factors = 6+1 = 7.
k² = x¹², with the result that the number of factors = 12+1 = 13.
According to Rule 2, the 7 factors that divide k must also divide k².
Thus:
Total number of positive integers that divide k² but not k = (total number of factors of k²) - (total number of factors of k) = 13-7 = 6.
Case 2: √k = x¹y¹, with the result that the number of factors = (1+1)(1+1) = 4
Since √k = x¹y¹, we get:
k = x²y², with the result that the number of factors = (2+1)(2+1) = 9.
k² = x�y�, with the result that the number of factors = (4+1)(4+1) = 25.
According to Rule 2, the 9 factors that divide k must also divide k².
Thus:
Total number of positive integers that divide k² but not k = (total number of factors of k²) - (total number of factors of k) = 25-9 = 16.
Since the answer to the question stem can be different values, INSUFFICIENT.
The correct answer is A.
