Last visit was: 23 Nov 2024, 22:16 It is currently 23 Nov 2024, 22:16

Close

GRE Prep Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GRE score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Own Kudos [?]: 12196 [21]
Given Kudos: 136
Send PM
Most Helpful Community Reply
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Own Kudos [?]: 12196 [11]
Given Kudos: 136
Send PM
General Discussion
avatar
Retired Moderator
Joined: 20 Apr 2016
Posts: 1307
Own Kudos [?]: 2273 [5]
Given Kudos: 251
WE:Engineering (Energy and Utilities)
Send PM
Verbal Expert
Joined: 18 Apr 2015
Posts: 30016
Own Kudos [?]: 36367 [0]
Given Kudos: 25928
Send PM
Re: Positive integer N has k positive divisors, and k has x posi [#permalink]
Expert Reply
Amazing explanation Sir !!

Regards
avatar
Intern
Intern
Joined: 29 Dec 2021
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 2
Send PM
Re: Positive integer N has k positive divisors, and k has x posi [#permalink]
pranab223 wrote:
GreenlightTestPrep wrote:
Positive integer N has k positive divisors, and k has x positive divisors. If k and x are odd integers greater than 2, what is the least possible value of N - k - x?


Show: ::
24



I normally choose numbers to replace unknown quantity for easy calculation

Points to remember

k has to be odd number, so that x becomes odd

Minimum possible value of \(k = 2^2 * 3^2 = 36\) , ( can't choose 1 because we need the divisors to be odd . \(1^1 = 1^2\) = doesn't make sense, so avoid)

therefore the positive divisors of 36 are = \(3*3 =9\)

Now \(9 = 3^2\)

therefore, the positive divisor of 9 =\(3\)

Now, possible value of \(N - k - x = 36 - 9 -3 = 24\)


Could you please explain how you came up with k=2^2 * 3^2 ? What is the idea behind giving that value to k? Also, the value of k is supposed to be odd. Why did you go with 36 as value of k?
Retired Moderator
Joined: 02 Dec 2020
Posts: 1831
Own Kudos [?]: 2146 [1]
Given Kudos: 140
GRE 1: Q168 V157

GRE 2: Q167 V161
Send PM
Positive integer N has k positive divisors, and k has x posi [#permalink]
1
\(k\) represents the no of divisors and which is supposed to be odd.

It should be that \(36\) will give us the smallest \(k\).

By taking \(2^2*3^2\), the \(k\) will be \(3*3 = 9 =\) no of divisor. We can't take odd power else the no of divisors will be even. Not possible in this case.

So when \(k=9\), the positive factors are \(3\) (1,3 and 9) so \(x = 3\)

So these are the smallest values.

Please ask if the doubt remains.

Gregaurangi wrote:
Could you please explain how you came up with k=2^2 * 3^2 ? What is the idea behind giving that value to k? Also, the value of k is supposed to be odd. Why did you go with 36 as value of k?
Intern
Intern
Joined: 05 Jan 2023
Posts: 22
Own Kudos [?]: 16 [2]
Given Kudos: 35
GRE 1: Q164 V160
Send PM
Re: Positive integer N has k positive divisors, and k has x posi [#permalink]
2
GreenlightTestPrep wrote:
Positive integer N has k positive divisors, and k has x positive divisors. If k and x are odd integers greater than 2, what is the least possible value of N - k - x?


Show: ::
24


There are three things to note that will make this solution easy.

1. Prime factorisation of a number will allow you to know the number of factors when you multiply the powers of the prime numbers after adding one to each of the powers.
2. Only prime numbers have a total number of factors that is odd.
3. A prime number can only be factored by 1 and itself and hence has two factors.

Going from the above,

A. N and k are prime numbers.
B. Hence if k and x are odd integers greater than 2, the smallest possible value of x is 3.
C. To find k, we have to figure out what perfect square will have only 3 positive factors. This can only be the perfect square of a prime number.
D. When we find k, we have to find the smallest possible number that has k positive factors.

Fo find k, we have to choose the smallest prime number that will be the factor of a perfect square that is still odd.

Let's try 2.

2 squared is 4. 4 is an even number and can't work.
3 squared is 9 which is an odd number so it works.

Hence k is 9

Smallest value of x=3
Smallest value of k=9
Smallest value of N=?

Let's remember that N is also a perfect square.

So let us look at the perfect squares above 9 from the smallest one that will give have 9 factors.

Let us note that since the perfect square will have 9 factors, we need something that meets statement 1 stated above will multiply to form 9.

Since all factors in a prime factorisation will have the lowest power of 1, the only way to achieve 9 is by getting a perfect square with two prime numbers that have powers of 2 giving

(2+1) * (2+1) according to the same statement 1 above.

Let's begin to check the prime factorisations of the perfect squares larger than 9.

4*4 = 16
Prime factorisation: 2 raised to power 4. It has (4+1)=5 factors and hence does not work.
5*5 = 25
Prime factorisation: 5 squared; It has (2+1)=3 factors and hence does not work.
6*6 = 36
Prime factorisation: 2 squared * 3 squared. It has (2+1)*(2+1) =9 factors

Hence the smallest value of N is 36

and N - k - x = 36 - 9 - 3 = 24


If you liked my explanation, please give me a kudos, thanks.

On another note, can someone please teach me how to use the symbols on this platform?
User avatar
GRE Prep Club Legend
GRE Prep Club Legend
Joined: 07 Jan 2021
Posts: 5043
Own Kudos [?]: 74 [0]
Given Kudos: 0
Send PM
Re: Positive integer N has k positive divisors, and k has x posi [#permalink]
Hello from the GRE Prep Club BumpBot!

Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Prep Club for GRE Bot
Re: Positive integer N has k positive divisors, and k has x posi [#permalink]
Moderators:
GRE Instructor
84 posts
GRE Forum Moderator
37 posts
Moderator
1111 posts
GRE Instructor
234 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne