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If \(k\) is a positive integer greater than 1, how many positive factors does \(k\) have?
(1) \(\frac{k}{11}\) is an integer which does not have a factor p such that \(1 < p < \frac{k}{11}\).
(2) The number of positive factors of \(k\) is odd.
(1) \(\frac{k}{11}\) is an integer which does not have a factor p such that \(1 < p < \frac{k}{11}\).
(2) The number of positive factors of \(k\) is odd.
30 Dec 2023, 22:38
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Official Solution:
If \(k\) is a positive integer greater than 1, how many positive factors does \(k\) have?
(1) \(\frac{k}{11}\) is an integer which does not have a factor p such that \(1 < p < \frac{k}{11}\).
This statement implies that \(\frac{k}{11}\) 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
If \(k\) is a positive integer greater than 1, how many positive factors does \(k\) have?
(1) \(\frac{k}{11}\) is an integer which does not have a factor p such that \(1 < p < \frac{k}{11}\).
This statement implies that \(\frac{k}{11}\) 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
