GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 18 Oct 2019, 07:35 GMAT 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

every week, we’ll send you an estimated GMAT 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.  Integer X represents the product of all integers between

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Intern  Joined: 29 Dec 2012
Posts: 16
Integer X represents the product of all integers between  [#permalink]

Show Tags

1
24 00:00

Difficulty:   55% (hard)

Question Stats: 56% (01:24) correct 44% (01:39) wrong based on 216 sessions

HideShow timer Statistics

Integer X represents the product of all integers between 1 to 25 (inclusive). The smallest prime factor of (x+1) must be

A. Between 1 to 10
B. Between 11 to 15
C. Between 15 to 20
D. Between 20 to 25
E. Greater than 25

Originally posted by schittuluri on 06 Aug 2014, 22:58.
Last edited by Bunuel on 13 Aug 2014, 11:26, edited 1 time in total.
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

Intern  Joined: 29 Oct 2013
Posts: 18
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

4
3
*The principle here is that two consecutive number don't have any factors in common but 1 and x and x+1 are obviously consecutive.

*So the smallest prime factor of 25!+1 is the smallest prime number following the largest prime factor of 25! i.e-23, which means this should be at least 29(not that it is)

Therefore, Answer is E
General Discussion
SVP  Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1749
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

4
1
schittuluri wrote:
Integer X represents the product of all integers between 1 to 25 (inclusive). The smallest prime factor of (x+1) must be _____________

A) Between 1 to 10
B) Between 11 to 15
C) Between 15 to 20
D) Between 20 to 25
E) Greater than 25

Answer = E = Greater than 25

This problem is asking smallest prime factor of (25!+1)

25! already have there prime factors 2,3,5,7,11,13.......... so on upto 23 (1 cannot be considered prime factor)

Just adding 1 to 25! will remove all the factors stated above;

so the smallest possible prime factor has to be greater than 25

Kindly update the OA
_________________
Kindly press "+1 Kudos" to appreciate Intern  Joined: 26 Feb 2015
Posts: 3
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

I still can't understand this question, can someone elaborate on this please?
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15271
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

2
1
Hi All,

This question is essentially just a 'clone' of the following one:
----------------------------
For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is:

1. between 2 and 10
2. between 10 and 20
3. between 20 and 30
4. between 30 and 40
5. greater than 40
-----------------------------

It's based on the exact same principals; the main idea though is:

"The ONLY number that will divide into X and X+1 is 1."

In other words, NONE of the factors of X will be factors of X+1, EXCEPT for the number 1.

Here are some examples:
X = 2
X+1 = 3
Factors of 2: 1 and 2
Factors of 3: 1 and 3
ONLY the number 1 is a factor of both.

X = 9
X+1 = 10
Factors of 9: 1, 3 and 9
Factors of 10: 1, 2, 5 and 10
ONLY the number 1 is a factor of both.
Etc.

Knowing this....we can deduce....
1) 25! will have LOTS of different factors
2) NONE of those factors will divide into 25! + 1.

25! contains all of the primes from 2 through 23, inclusive, so NONE of those will be in 25! + 1. We don't even have to calculate which prime factor is smallest in 25! + 1; we know that it MUST be a prime greater than 23....and there's only one answer that fits.

GMAT assassins aren't born, they're made,
Rich
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

EMPOWERgmatRichC wrote:
Hi All,

This question is essentially just a 'clone' of the following one:
----------------------------
For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is:

1. between 2 and 10
2. between 10 and 20
3. between 20 and 30
4. between 30 and 40
5. greater than 40
-----------------------------

It's based on the exact same principals; the main idea though is:

"The ONLY number that will divide into X and X+1 is 1."

In other words, NONE of the factors of X will be factors of X+1, EXCEPT for the number 1.

Here are some examples:
X = 2
X+1 = 3
Factors of 2: 1 and 2
Factors of 3: 1 and 3
ONLY the number 1 is a factor of both.

X = 9
X+1 = 10
Factors of 9: 1, 3 and 9
Factors of 10: 1, 2, 5 and 10
ONLY the number 1 is a factor of both.
Etc.

Knowing this....we can deduce....
1) 25! will have LOTS of different factors
2) NONE of those factors will divide into 25! + 1.

25! contains all of the primes from 2 through 23, inclusive, so NONE of those will be in 25! + 1. We don't even have to calculate which prime factor is smallest in 25! + 1; we know that it MUST be a prime greater than 23....and there's only one answer that fits.

GMAT assassins aren't born, they're made,
Rich

That question is discussed here: for-every-positive-even-integer-n-the-function-h-n-is-126691.html

Rich, can you please post this solution to that thread? Thank you!
_________________
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15271
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

Hi Bunuel,

Done! I've also adjusted the explanation to fit that thread/question.

GMAT assassins aren't born, they're made,
Rich
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

EMPOWERgmatRichC wrote:
Hi Bunuel,

Done! I've also adjusted the explanation to fit that thread/question.

GMAT assassins aren't born, they're made,
Rich

__________
Thank you!
_________________
Senior Manager  Status: Math is psycho-logical
Joined: 07 Apr 2014
Posts: 402
Location: Netherlands
GMAT Date: 02-11-2015
WE: Psychology and Counseling (Other)
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

EMPOWERgmatRichC wrote:
Hi Bunuel,

Done! I've also adjusted the explanation to fit that thread/question.

GMAT assassins aren't born, they're made,
Rich

I have a question though. If for example we take the number 14 and 15, which are consecutive, we can see that 14 = 2 * 7 and 15 = 3 * 5.

Indeed they do not have any factors in common (except for 1). However, even though 15 is following 14, its smallest prime factor (3) is not the one that follows 14's greatest prime factor (7).

Based on the principle above, shouldn't 15 have 11 as its lowest prime factor?
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15271
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

Hi pacifist85,

I used individual values to prove the following point: If we're dealing with integers, then a positive integer that divides into X will NOT divide into (X+1). The exception is the number 1.

This prompt asks us to deal with 25 FACTORIAL (not 25), so we have to take the above point and apply it here on a larger scale.....

Any positive integer (except 1) that divides into 25! will NOT divide into (25! + 1).

Since 14 is "part" of the product of 25!, then the 2 and the 7 (that "make up" the 14) will divide into 25!. By extension, they will NOT divide into 25!+1.

EVERY prime number up to (and including) 23 are a part of the product of 25!, so they will ALL divide into 25!. By extension, they will NOT divide into 25!+1.

To be honest, I can't tell you what the smallest prime number is that divides into 25!+1, but since the question did not ask me for THAT answer, I'm not going to worry about it. I just know that it's a prime number greater than 23. Based on the way the answer choices are written, there's only one answer that makes sense....

GMAT assassins aren't born, they're made,
Rich
_________________
Senior Manager  Status: Math is psycho-logical
Joined: 07 Apr 2014
Posts: 402
Location: Netherlands
GMAT Date: 02-11-2015
WE: Psychology and Counseling (Other)
Integer X represents the product of all integers between  [#permalink]

Show Tags

EMPOWERgmatRichC wrote:
Hi pacifist85,

I used individual values to prove the following point: If we're dealing with integers, then a positive integer that divides into X will NOT divide into (X+1). The exception is the number 1.

This prompt asks us to deal with 25 FACTORIAL (not 25), so we have to take the above point and apply it here on a larger scale.....

Any positive integer (except 1) that divides into 25! will NOT divide into (25! + 1).

Since 14 is "part" of the product of 25!, then the 2 and the 7 (that "make up" the 14) will divide into 25!. By extension, they will NOT divide into 25!+1.

EVERY prime number up to (and including) 23 are a part of the product of 25!, so they will ALL divide into 25!. By extension, they will NOT divide into 25!+1.

To be honest, I can't tell you what the smallest prime number is that divides into 25!+1, but since the question did not ask me for THAT answer, I'm not going to worry about it. I just know that it's a prime number greater than 23. Based on the way the answer choices are written, there's only one answer that makes sense....

GMAT assassins aren't born, they're made,
Rich

Thank you Rich, I understood that.

But I read in one of the posts above that it makes sense that it is going to be large than 23, because the greatest prime factor of 25! is 23. So, for 25!+1 the lowest prime factor should be higher than 23.

In the same logic, why isn't this true for 14 and 15? So, why isn't the smallest prime factor of 15 larger than 7, which is the greatest prime factor of 14?

So, I understood why the smallest prime will be not one of the primes of 25!. What I couldn't understand is the principle that it must be higher than 23, because I thought that the principle said that for 2 consecutive numbers the lowest prime of the second should be higher than the highest prime of the first, which is not true. But I think that I get it now.

The point is that 25!+1 cannnot have the same prime factors are 25!. And 25! has all the prime factors up to that point. So, this is why 25!+1 must have as its lowest factors a number larger than 23, because it cannot share any of the prime factors of 25! and 25! has all the primes from 1 up to 25, 23 including.
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15271
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

Hi pacifist85,

From your post, it looks like you understand the individual points/ideas, but you're not connecting them (completely).

The first point is that the prime factors of X will NOT divide into (X+1). The prime factors that WILL divide into (X+1) might be bigger OR smaller than the ones that divide into X, but that was never a part of the discussion.

25! = (1)(2)(3)(4)(5)...(7)....(11)...(13)...(17)...(19)...(23)(24)(25), so NONE of those factors will divide into (25!+1). Since that big product includes EVERY prime from 2 through 23 (inclusive), NONE of those primes will divide into (25!+1). Thus, the prime factors that WILL divide into (25!+1) MUST be greater than 23 because they're the only primes that are left.

GMAT assassins aren't born, they're made,
Rich
_________________
Current Student Joined: 18 Oct 2014
Posts: 803
Location: United States
GMAT 1: 660 Q49 V31 GPA: 3.98
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

1
schittuluri wrote:
Integer X represents the product of all integers between 1 to 25 (inclusive). The smallest prime factor of (x+1) must be

A. Between 1 to 10
B. Between 11 to 15
C. Between 15 to 20
D. Between 20 to 25
E. Greater than 25

25!+1 is consecutive to 25!

Two consecutive numbers don't have any common factor other than 1. Hence there are no factors from 2-25 that are in 25!+1.

Hence, smallest prime factor is greater than 25

E is the answer
_________________
I welcome critical analysis of my post!! That will help me reach 700+
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3074
Re: Integer X represents the product of all integers between  [#permalink]

Show Tags

Solution

Given
• X = the product of all integers between 1 to 25

To find
• The smallest prime factor of x +1

Approach and Working out
• X = 1× 2 × 3×….×25 = 25!
• X + 1 = 25! +1
• The prime factor of X + 1 will completely divide 25! + 1. So, the remainder will be 0.

Now, If 25! + 1 is divided by any prime number between 1 and 25 then:
• 25! will leave 0 as the remainder.
• 1 will leave 1 as the remainder.
o So, 1 will be the remainder when 25! + 1 is divided by any prime number between 1 and 25.

Hence, the prime factor of 25! + 1 is certainly greater than 25.

Thus, option E is the correct answer.

Correct Answer: Option E
_________________ Re: Integer X represents the product of all integers between   [#permalink] 29 Sep 2019, 05:47
Display posts from previous: Sort by

Integer X represents the product of all integers between

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

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