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

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Source: — Data Sufficiency |

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by [email protected] » Sun Aug 25, 2013 3:57 pm
Hi guerrero,

For this DS question, you need to think about what's possible and what's not (by coming up with examples).

I'm going to give you a hint that will make finding those examples a lot easier, then let you try this question again.

At the beginning, we're told that K is a POSITIVE INTEGER. In Fact 1, we're told that 2 does NOT evenly divide into K. That means that K CANNOT BE EVEN. Now, the only numbers that are possible are odd. Playing with THAT info, can you figure out if K is prime?

Once you've dealt with Fact 1, you'll use a similar approach for Fact 2.

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by ganeshrkamath » Sun Aug 25, 2013 10:03 pm
guerrero wrote:If k is a positive integer, is k a prime number?

(1) No integers between 2 and square root of k, inclusive divides k evenly.
(2) No integers between 2 and k/2 divides k evenly, and k is greater than 5.

OA D
Statement 1:
Property of a prime number: k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.
Sufficient

Statement 2:
k/2 >= sqrt(k)
So it satisfies the above-mentioned property.
Sufficient

Choose D

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by Uva@90 » Wed Sep 04, 2013 9:23 pm
ganeshrkamath wrote:
guerrero wrote:If k is a positive integer, is k a prime number?

(1) No integers between 2 and square root of k, inclusive divides k evenly.
(2) No integers between 2 and k/2 divides k evenly, and k is greater than 5.

OA D
Statement 1:
Property of a prime number: k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.
Sufficient

Statement 2:
k/2 >= sqrt(k)
So it satisfies the above-mentioned property.
Sufficient

Choose D

Cheers
Hi Ganeshrkamath,

I am confused with your solution for the second statement.

The property of prime number says,
k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.

But K/2 >= sqrt(K) which may not be in the range of 2 and sqrt(k). Then, how does it satisfies the above property ?

Please clarify it...

Thanks in advance.

Regards,
Uva.
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by ganeshrkamath » Wed Sep 04, 2013 9:53 pm
Uva@90 wrote:
ganeshrkamath wrote:
guerrero wrote:If k is a positive integer, is k a prime number?

(1) No integers between 2 and square root of k, inclusive divides k evenly.
(2) No integers between 2 and k/2 divides k evenly, and k is greater than 5.

OA D
Statement 1:
Property of a prime number: k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.
Sufficient

Statement 2:
k/2 >= sqrt(k)
So it satisfies the above-mentioned property.
Sufficient

Choose D

Cheers
Hi Ganeshrkamath,

I am confused with your solution for the second statement.

The property of prime number says,
k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.

But K/2 >= sqrt(K) which may not be in the range of 2 and sqrt(k). Then, how does it satisfies the above property ?

Please clarify it...

Thanks in advance.

Regards,
Uva.
Since sqrt(k) <= k/2, no factor lies between 2 and k/2 implies that no factor lies between 2 and sqrt(k).

No factors in the range [2. k/2] is the same as no factors in the range [2, sqrt(k)] and no factors in the range [sqrt(k), k/2].

Hope this helps.
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by vinay1983 » Thu Sep 05, 2013 1:22 am
Ho! Everything said till now is going overhead. Can me somebody explain this ina simple way!

Thanks!
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by Uva@90 » Thu Sep 05, 2013 2:13 am
vinay1983 wrote:Ho! Everything said till now is going overhead. Can me somebody explain this ina simple way!

Thanks!
vinay1983,

Which part you didn't understand ??

Regards,
Uva.
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by vinay1983 » Thu Sep 05, 2013 2:20 am
Uva@90 wrote:
vinay1983 wrote:Ho! Everything said till now is going overhead. Can me somebody explain this ina simple way!

Thanks!
vinay1983,

Which part you didn't understand ??

Regards,
Uva.
Sadly everything excluding the question :cry:
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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by Uva@90 » Thu Sep 05, 2013 3:32 am
vinay1983 wrote:
Uva@90 wrote:
vinay1983 wrote:Ho! Everything said till now is going overhead. Can me somebody explain this ina simple way!

Thanks!
vinay1983,

Which part you didn't understand ??

Regards,
Uva.
Sadly everything excluding the question :cry:
Fine!!!

To find: Whether k is prime or not ?

Before diving into answer remember this rule,

"Trial division

The most basic method of checking the primality of a given integer n is called trial division.
This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n.
If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime." Source = Wikipedia.

In simple it states
k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.

Let's move to the problem now,

Statement 1: It implies that K is not divisible by any number between 2 and sqrt(k).
Hence, according to the above rule K is prime Number.
A--sufficient.

Statement 2:
Half of any number is always greater than or equal to the sqrt of any number.
So,
K/2 >= SQRT(K)
From the statement 2 we can understand K is not divisible by any number between 2 and k/2.
since sqrt(k) is less than or equal to k/2.
We can re-write statement as ,
K is not divisible by any number between 2 and sqrt(k).

Which is same as statement 1.
Hence Sufficient.

OA is D

Hope I am clear.

Regards,
Uva.
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by vinay1983 » Thu Sep 05, 2013 4:24 am
Uva@90 wrote:
vinay1983 wrote:
Uva@90 wrote:
vinay1983 wrote:Ho! Everything said till now is going overhead. Can me somebody explain this ina simple way!

Thanks!
vinay1983,

Which part you didn't understand ??

Regards,
Uva.
Sadly everything excluding the question :cry:
Fine!!!

To find: Whether k is prime or not ?

Before diving into answer remember this rule,

"Trial division

The most basic method of checking the primality of a given integer n is called trial division.
This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n.
If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime." Source = Wikipedia.

In simple it states
k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.

Let's move to the problem now,

Statement 1: It implies that K is not divisible by any number between 2 and sqrt(k).
Hence, according to the above rule K is prime Number.
A--sufficient.

Statement 2:
Half of any number is always greater than or equal to the sqrt of any number.
So,
K/2 >= SQRT(K)
From the statement 2 we can understand K is not divisible by any number between 2 and k/2.
since sqrt(k) is less than or equal to k/2.
We can re-write statement as ,
K is not divisible by any number between 2 and sqrt(k).

Which is same as statement 1.
Hence Sufficient.

OA is D

Hope I am clear.

Regards,
Uva.
Yes teacher (Today is teacher's day!) i think i am able to understand it.

:)
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by sanju09 » Sat Sep 14, 2013 3:39 am
guerrero wrote:If k is a positive integer, is k a prime number?

(1) No integers between 2 and square root of k, inclusive divides k evenly.
(2) No integers between 2 and k/2 divides k evenly, and k is greater than 5.

OA D
I. Let's test back the statement as if k for a prime; does it hold true? If k is 29, its square-root is a bit more than 5. How much more than 5? We least bother about what we don't have to use in the question. And none of 2, 3, 4, or 5 divides 29 evenly. Few doubts still in mind! Is this because k is prime, will it happen with all primes, a new rule if any, who cares, we have more types of numbers than primes for k to test back the statement. We've 65 waiting, we know its square-root is little more than 8, and we've this 5 among 2, 3, 4, 5, 6, 7, and 8 that divides 65 evenly. Hmm...alright, what if k is 121, its square-root is a prime accidently, so we got another time on primes, no number among the list EXCEPT 11 itself divides 121 evenly. It may mean because 121 is composite, so happened this, who knows? Any bigger prime in mind to pass the test? We see composites are failing the statement test and primes passing. Do we know a category other than primes and composites in positive integers? Yeah it's 1.

OK, one question, when we say between a and b, can we say between b and a is same as that? I personally think I can because I believe in a philosophy that tells me that distance between you and me is same as distance between me and you unless specified.


How D? How are you?

No number among 2 and 1 can divide 1 evenly, is it really so happening here? Is 1 a prime? Which part of the question plus this statement validates k is not 1? Does it say that between 2 and a positive number means 2 or more only? Why not 2 or less, if it's possible and available too? I doubt the source somehow.

It's the other statement that tells us k is not 1, but how is it helpful in the first one I can't understand.

GMAT expects us to work out with numbers as suitable examples if a must know rule is missing while solving questions testing us over the properties of integers we must know to crack the questions there.

It's a good number theory question missing just few things to match the standard of GMAT where they leave nothing to assume.
The mind is everything. What you think you become. -Lord Buddha



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