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

Question

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

ganand · Accepted Answer

@

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.

chetan2u · Answer

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

stonecold · Answer

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

chetan2u · Answer

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..

JS1290 · Answer

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!

matthewsmith_89 · Answer

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

niks18 · Answer

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

JS1290 · Answer

Thank you so much for explaining it to me. Makes complete sense now! Kudos given to both ganand & niks18!

adityasuresh · Answer

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)

pratiksha1998 · Answer

Can someone explain how to solve this?

Regor60 · Answer

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

ruslanismayil · Answer

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!

shadow04 · Answer

Agreed. Typo: I think you meant 40! and not 20!.

We know that   11  ,   19  , and   37   are prime factors of 40! because 40! means:
= 40! = 40 x 39 x 38 x   37   x 36 x 35 x 34 x 32 x 31 ...   19   x 18 x 17 x 16 x 15 x 14 x 13 x 12 x   11   x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.

Basically, if the answer choices given were any integer less than 40, it would be considered a factor of 40! since 40! is the product of every number below it. Also, since the question says " prime " factor, make sure that 11, 19, and 37 are prime, which they are.

KyleOlive · Answer

Another trick I used that helped me was that 40! will have multiple trailing zeros since it has multiple factors of 0. Consequently, 40!+1 will be a number x...xx0000001. Multiple of 11, 19, 37 to my knowledge would be unlikely end in 00001, so my best guess was none.

findingmyself · Answer

Making this simple and easy:

40! as a number will end with a zero
40!+1 will have 1 in unit digit

All the answer choices can give 1 in unit digit when multiplied by an integer between 0-9

11*1=11 (unit digit 1)
19*9=xx1(unit digit 1)
37*3=xx1(unit digit 1)

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

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GMAT Problem Solving (PS) Questions

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

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