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C
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37% (02:07) correct
63%
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wrong
based on 233
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12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun
If k is a positive integer, how many positive factors does k have?
(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.
(2) The number of positive factors of k is odd.
If k is a positive integer, how many positive factors does k have?
(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.
(2) The number of positive factors of k is odd.
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16 Dec 2023, 06:58
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GMAT Club Official Explanation:
If k is a positive integer, how many positive factors does k have?
(1) k/11 is an integer which does not have a factor p such that 1 < p < k/11.
This statement implies that k/11 is a prime number, as a prime number only has two factors: 1 and itself, and no other factors between 1 and the number itself. Therefore, k = 11*prime. If this prime is any prime other than 11, then k has (1 + 1)(1 + 1) = 4 factors. However, if this prime is 11, then k = 11^2, and the number of factors is (2 + 1) = 3. Not sufficient.
(2) The number of positive factors of k is odd.
Only squares of integers have an odd number of factors, while all other integers have an even number of factors. Therefore, this statement just implies that k = integer^2. Not sufficient.
(1)+(2) Since from (2), k must be the square of an integer, then from (1) it follows that it must be 11^2, giving the number of factors equal to 3. Sufficient.
Answer: C.
General Discussion
12 Dec 2023, 07:50
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Bunuel
(1) k/11 is an integer which does not have a factor p such that 1 < p < k.
k/11 is a prime number.
Hence, k can be 11 * 11. In this case the number of factors = 3
or
k can be 11 * some other prime number. In this case the number of factors = 4
The statement alone is not sufficient.
(2) The number of positive factors of k is odd.
The statement alone is not sufficient. k can be any odd number.
Statement (1) + Statement (2)
The statements combined tell us the number of factor is odd. Out of two possibilities in Statement 1, we can eliminate the possibility in which the number of factor is 4. Hence, the number of factor is 3.
The statements combined are sufficient.
IMO C
