NandishSS wrote:If k is a positive integer, is k a prime number?
(1) No integers between 2 and k√k, inclusive divides k evenly.
(2) No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
We are given that k is a positive integer and need to determine whether k is prime. Recall the following:
If no integers between 2 and √k, inclusive, divide k evenly, then k is a prime.
For example, 17 is a prime since none of the integers 2, 3, and 4 (notice that √17 ≈ 4.1) divide 17 evenly.
Statement One Alone:
No integers between 2 and k√k, inclusive divides k evenly.
Since no integers between 2 and k√k, inclusive divide k evenly, it must be true that no integers between 2 and √k (notice that √k < k√k), inclusive, divide k evenly. So k must be a prime. Statement one alone is sufficient.
Statement Two Alone:
No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
Since no integers between 2 and k/2, inclusive, divide k evenly, it must be true that no integers between 2 and √k (notice that √k < k/2 when k > 5), inclusive, divide k evenly. So k must be a prime. Statement two alone is also sufficient.
Answer: D
