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If q = 40! + 1, which of the following cannot be a prime factor of q?

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Bunuel
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If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   29 Apr 2016, 16:09
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If q = 40! + 1, which of the following cannot be a prime factor of q?

I. 11
II. 19
III. 37

A. I​ only
B. III only
C. II and III
D. I and II
E. I​, II, and III
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E
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ganand
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If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   Updated on: 23 Apr 2017, 21:52
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csaluja wrote:
Hi,

I am still unable to understand how 11, 19, and 37 are not prime factors of 40! +1. Can anyone please explain it to me in a different way?

Thanks!


Hi csaluja,

I'll try to explain it. Let's understand the definition of co-prime numbers:

Definition: A set of integers can also be called co-prime if its elements share no common positive factor except 1.

For example, consider 15 and 22. Factors of 15 are 1, 3, 5, and 15. Factors of 22 are 1, 2, 11, and 22.
Hence, 15 and 22 are co-prime numbers.

Now consider 15 and 21. Factors of 21 are 1, 3, 7, and 21. 15 and 21 have two common factors 1 and 3. Hence, not a co-prime numbers.


Result: Any two consecutive integers are always co-prime numbers.

40! and 40!+1 are consecutive integers. Hence, no common factors. => 11, 19, and 37 are factors of 40!. So, these can't be a factor of 40!+1.

Now give a try to following GMAT Prep question:

for-every-positive-even-integer-n-the-function-h-n-is-defined-to-be

Hope it helps.

Originally posted by ganand on 21 Apr 2017, 19:36.
Last edited by ganand on 23 Apr 2017, 21:52, edited 1 time in total.
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If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   29 Apr 2016, 19:39
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Bunuel wrote:
If q = 40! + 1, which of the following cannot be a prime factor of q?

I. 11
II. 19
III. 37

A. I​ only
B. III only
C. II and III
D. I and II
E. I​, II, and III


HI,
we should remember that any factorial + 1 is co-prime to all numbers less than the factorial integer, including itself, and thus does not have any factors except 1
WHY?- Because factorial is the product of all the numbers till that factorial so when you add 1 to that number, all numbers will give a remainder of 1..

all the numbers 11, 19, and 37 are smaller than 40, so none of them will be factors of 40!+1..
ans E
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stonecold
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   02 May 2016, 17:13
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chetan2u wrote:
Bunuel wrote:
If q = 40! + 1, which of the following cannot be a prime factor of q?

I. 11
II. 19
III. 37

A. I​ only
B. III only
C. II and III
D. I and II
E. I​, II, and III


HI,
we should remember that any factorial + 1 is a prime number, and thus does not have any factors except 1
WHY?- Because factorial is the product of all the numbers till that factorial so when you add 1 to that number, all numbers will give a remainder of 1..

all the numbers 11, 19, and 37 are smaller than 40, so none of them will be factors of 40!+1..
ans E

I disagree on one thing here =>
You said it cannot have any factors other than one
How about 43 ? or 93 ? or any number greater than 40?
What you wrote is fine for numbers<40
But the theory cannot be used for numbers greater than 40..!
we cannot be sure here as => non multiple + non multiple = may be a multiple or a non multiple
How about using the rule => Multiple +multiple = multiple
multiple + non multiple = non multiple

Lemme know if i am missing something here
Regards
StoneCold
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chetan2u
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   02 May 2016, 19:00
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stonecold wrote:
chetan2u wrote:
Bunuel wrote:
If q = 40! + 1, which of the following cannot be a prime factor of q?

I. 11
II. 19
III. 37

A. I​ only
B. III only
C. II and III
D. I and II
E. I​, II, and III


HI,
we should remember that any factorial + 1 is a prime number, and thus does not have any factors except 1
WHY?- Because factorial is the product of all the numbers till that factorial so when you add 1 to that number, all numbers will give a remainder of 1..

all the numbers 11, 19, and 37 are smaller than 40, so none of them will be factors of 40!+1..
ans E

I disagree on one thing here =>
You said it cannot have any factors other than one
How about 43 ? or 93 ? or any number greater than 40?
What you wrote is fine for numbers<40
But the theory cannot be used for numbers greater than 40..!
we cannot be sure here as => non multiple + non multiple = may be a multiple or a non multiple
How about using the rule => Multiple +multiple = multiple
multiple + non multiple = non multiple

Lemme know if i am missing something here
Regards
StoneCold


Hi,
It is related to common factors between that factorial +1 and all other numbers <40.
and it is not concerned with higher number than the factorial..
example 4!+1 = 24+1 = 25 it is div by 5..
similarily 5!+1 is div by 11 but no other number <5 , as it is co-prime with others..

But yes my post does convey that it is prime rather than co-prime..
Thanks..
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matthewsmith_89
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   21 Apr 2017, 20:53
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csaluja wrote:
Hi,

I am still unable to understand how 11, 19, and 37 are not prime factors of 40! +1. Can anyone please explain it to me in a different way?

Thanks!


Hi,

Let's say Q = 4! + 1

Q = 4 x 3 x 2 x 1 +1
Q = 24 + 1
Q = 25
25 is not divisible by 2, 3 and 4
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   23 Apr 2017, 12:00
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csaluja wrote:
Hi,

I am still unable to understand how 11, 19, and 37 are not prime factors of 40! +1. Can anyone please explain it to me in a different way?

Thanks!


Hi,

Any two consecutive integer will not have a common prime factor, for example, 2&3, 6&7, 104 (prime factors - 2&13) & 105 (prime factors - 3,5&7) and so on. you can experiment with other numbers as well. These numbers are co-prime (explained in earlier posts)

Now, let 40!=x ; so, 40!+1 = x+1.
Thus x & x+1 are consecutive numbers. So both will not have any common prime factor.
Since 11,19 & 37 are prime factors of 40! so they will not be the prime factors of 40!+1
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   16 Jun 2022, 10:19
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pratiksha1998 wrote:
Can someone explain how to solve this?



This is easier than has been discussed.

Call a potential factor N.

That would mean:

(40!+1)/N would yield an integer result.

The above can be rewritten as:

40!/N + 1/N

All of the answer choices are embedded in the factorial of 40!, so that part yields an integer.

But 1/N also has to yield an integer to make the sum an integer.

The only N that would work is 1, but that's not among the answer choices.

Posted from my mobile device
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   21 Apr 2017, 18:19
Hi,

I am still unable to understand how 11, 19, and 37 are not prime factors of 40! +1. Can anyone please explain it to me in a different way?

Thanks!
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   23 Apr 2017, 16:32
Thank you so much for explaining it to me. Makes complete sense now! Kudos given to both ganand & niks18!
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   09 May 2022, 22:56
If q = 40! + 1, which of the following cannot be a prime factor of q?

I. 11
II. 19
III. 37

Hi kindly let me know if this approach is right for this type of questions. I used distributive property.

\frac{A+B}{C} =\frac{ A}{C + B/C }

For 11 : \frac{40! + 1}{11 = 40!/11 + 1/11} - \frac{1}{11} is not an integer, hence its not a factor of Q.

Similarly for 19 : \frac{40!+1}{19 } = \frac{40}{19} which is an integer + \frac{1}{19} not an integer, hence not a factor of Q

Similarly for 37 :\frac{ 40!+1}{37} = \frac{40!}{37 }which is an integer + \frac{1}{37} not an integer.
Hence not a factor of Q .

OA : (E)
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   16 Jun 2022, 08:36
Can someone explain how to solve this?
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   21 Jan 2023, 12:05
I dont think the problem here is not the rule of consecutive prime numbers for people, but whether 11, 19 and 37 are prime factors of 20! :)
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink] New post   29 Feb 2024, 10:52
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Re: If q = 40! + 1, which of the following cannot be a prime factor of q? [#permalink]
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