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If k is a positive integer, is √k an integer?
In other words, the question is asking is k the square of an integer?
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(1) The number of positive factors of k is a prime number
This means k has either 2 factors or an odd number of factors.
Only prime numbers have 2 factors. In that case, k isn't the square of a number and √k isn't an integer.
If k has odd number of factors, it means k is the even power of a number. Which means, k is also the square of a number and √k is an integer.
Hence, (1) is not sufficient
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(2) k has a factor p such that 1 < p < k
This means that k has more than 2 factors. So, it has 3 or more factors. (Now, I don't know how literally we should take "a factor". Does it mean "exactly one" or "at least one"? If "exactly one", then it means k has exactly 3 factors, meaning k is the square of a prime number, and statement 2 will be sufficient. But, I chose to go with the 2nd interpretation, i.e. "k has at least 1 factor between 1 and k")
6 has at least 1 factor between 1 and 6, and it's not a square
9 has at least 1 factor between 1 and 9, and it is a square.
So, clearly statement (2) isn't sufficient either.
Taking (1) and (2) together:
We know k doesn't have 2 factors. It has more than 2 factors. So, it can have only odd number of factors.
As discussed under (1), if it has odd number of factors, it means that k is the even power of an integer. That means, k is a square and √k is an integer.
(1) and (2) are together sufficient to answer the question.
Answer: C