bhandariavi wrote:
If k is a positive integer, is k a prime number?
(1) No integers between 2 and \(\sqrt{k}\), inclusive divides k evenly.
(2) No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
Target question: Is k a prime number?The key property here addresses how many numbers we must check in order to confirm a number is prime. For example, let's say we want to determine whether 23 is a prime number.
We first ask, "Is 23 divisible by 2?" NO.
Then we ask, "Is 23 divisible by 3?" NO.
"Is 23 divisible by 4?" NO.
"Is 23 divisible by 5?" NO.
.
.
.
In order to determine whether 23 is prime, must we keep testing every value up to 22?
The answer is no; we need only test the greatest integer that's less than or equal to √23
√23 = 4.something
So once we have confirmed that 23 is not divisible by 2, 3 or 4, we can stop testing values and conclude that 23 must be prime
In general, we can say:
If positive integer n has no divisors from 2 to √n inclusive, then n is prime With that in mind let's take a look at the statements....
Statement 1: No integers between 2 and \(\sqrt{k}\), inclusive divides k evenly.In other words, positive integer k has no divisors from 2 to √n inclusive
By the above
property, we can conclude that
k must be primeSince we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.[/quote]This statement is very similar to statement 1
Notice that, if k is greater than 5, then k/2 is greater than √k
So, is this case, to determine whether k is prime, we're testing an even
wider range of values than we did with statement 1.
So if statement 1 is sufficient (by testing possible divisors from 2 to √k inclusive), then statement 2 must also be sufficient, since it tests and even wider range of possible divisors (from 2 to k/2 inclusive).
Answer: D
Cheers,
Brent