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# If k is a positive integer, is k a prime number?

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Re: Is k a prime number? [#permalink]
hie Bunuel,

wat does 'divides evenly' means here?
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Re: Is k a prime number? [#permalink]
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ramana wrote:
hie Bunuel,

wat does 'divides evenly' means here?

x divides k evenly, just means that x is a factor of k, divides k without leaving a remainder. For example, 5 divides 15 evenly --> 15/5=3.
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Re: k is a positive number. Is K a prime number? 1) No integers [#permalink]
D. Very helpful analysis by bunuel
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Re: If k is a positive integer, is k a prime number? [#permalink]
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.

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.

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Re: If k is a positive integer, is k a prime number? [#permalink]
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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 prime
Since 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).

Cheers,
Brent
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Re: If k is a positive integer, is k a prime number? [#permalink]
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Re: If k is a positive integer, is k a prime number? [#permalink]
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