Integer X represents the product of all integers between

Question

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

fra · Accepted Answer

*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

Bunuel · Answer

Similar questions to practice: for-every-even-positive-integer-m-f-m-represents-the-produ-168636.html for-every-positive-even-integer-n-the-function-h-n-is-126691.html for-every-positive-even-integer-n-the-function-h-n-149722.html for-any-integer-p-p-is-equal-to-the-product-of-all-the-int-112494.html x-is-the-product-of-all-even-numbers-from-2-to-50-inclusive-156545.html if-a-and-b-are-odd-integers-a-b-represents-the-product-of-144714.html the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html for-every-even-positive-integer-m-f-m-represents-165666.html Hope this helps.

PareshGmat · Answer

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

Answer = E

Kindly update the OA

Armond · Answer

I still can't understand this question, can someone elaborate on this please?

EMPOWERgmatRichC · Answer

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.

Final Answer:  E

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

Bunuel · Answer

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!

EMPOWERgmatRichC · Answer

Hi Bunuel,

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

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

Bunuel · Answer

__________
Thank you!

pacifist85 · Answer

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?

EMPOWERgmatRichC · Answer

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

pacifist85 · Answer

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.

EMPOWERgmatRichC · Answer

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

Divyadisha · Answer

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

EgmatQuantExpert · Answer

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

omer339 · Answer

Bunuel  , can you please explain the solution.... I am still unable to understand the approach. Thanks a lot!!!

EMPOWERgmatRichC · Answer

Hi omer339

This question is based around the following math rule:

"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.

In this question, we're told that X is the PRODUCT of the first 25 integers. This means X = 25!.... (25)(24)(23).... (3)(2)(1). We're asked for the SMALLEST prime factor of (X+1).

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

summerbummer · Answer

---------------------------

Translation is where I'm facing an issue. Can someone help?

Agent74 · Answer

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

Rule  - Consecutive numbers always have 1 as their common factor

X = Product of all integers between 1 to 25 so 25! = X

25! = {1*2*3*.............25} = say some number

Now, X +1 = 25! +1

So, as per rule above their common factor will be 1, however 1 is not a prime number (we have to find the smallest prime number)

Hence, their common factor will lie beyond 25!+1. So, E

bumpbot · Answer

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Integer X represents the product of all integers between

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Integer X represents the product of all integers between

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