HOT Competition 27 Aug/8PM: If k is a positive integer, is k^(1/2) an

If k is a positive integer, is an integer?

(1) The number of positive factors of k is a prime number
(2) k has a factor p such that 1 < p < k

1) if k has just 2 factors, then square root k is not an integer. for example: when k = 2 or 3.
But when k has 3 or 5 factors, then square root k is an integer. Example square root of 4 or 16. Not sufficient.

2) k is not a prime integer. Clearly not sufficient. Square root of 6 not an integer but square root of 9 is an integer. Not sufficient

Together, the number of factors of a square of an integer is always odd. And as we can infer the number not factors of k cannot be 2, so in every case square root k will be an integer. Sufficient

C is our answer.

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K is an integer \(\sqrt{k}\) is an integer or not.
We have to find if k is square of any number.
All the number has factor pairs except squares. They always have odd number of factors.

Statement 1:
Number of factors are prime number.
Except 2 all the prime numbers are odd.
So we have 2 options either k is square or k is prime(in that case it will have 2 factors) : insufficient

Statement 2:
p is a factor of k and it is between 1 and p so we know that k is not prime. But still we don't know if k is square: insuff

Combining 1+2
We know that k has odd number of factors.
So, C is sufficient.

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In other words, the question is asking is k the square of an integer?


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


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

If k is a positive integer, is √k an integer?

(1) The number of positive factors of k is a prime number- insufficient as k can 2, factors of 2 are 1,2. so (k)^1/2 is not an integer. k can be 4. factors of 4 are 1,2, & 4. so (k)^1/2 is an integer.
(2) k has a factor p such that 1 < p < k. insufficient as k can be 4 and the given condition will be true or k can be 8 and the given condition will not be true.

Together taking (1) & (2) k can be 4, 9, 16, 25 etc. so (k)^1/2 will always be an integer.
so c is sufficient.

ANswer: Option C



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