The main rule is that two consecutive integers cannot have the same prime factors.

For example 6 can be factorized 2 and 3 but 7 does not have factors 2 and 3.

Factors of 18! cannot be factors of 18!+1.

All the other options apart from C are factors of 18!.

Hence C is the only option possible.

Hi Guys,

I understand perfectly well the explanation. And, although is a little old the post, does the question need to be edited to: "Which of the following is not a factor of 18!+1?
Thanks in advance for anything.

In this case probably no.

Important Concept #1: If k > 1, and k is a factor (divisor) of N, then k is not a divisor of N+1
For example, since 7 is a factor of 350, we know that 7 is not a factor of (350+1)
Similarly, since 8 is a factor of 312, we know that 8 is not a factor of 313

Important Concept #2: If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
-----ONTO THE QUESTION!---------------------


We're going to use both concepts to ELIMINATE 4 of the 5 answer choices.
A. 15
Since 18! = (18)(17)(16)(15)....(3)(2)(1), we can see that 15 is a factor (divisor) of 18!
So, according to Concept #1, 15 is NOT a divisor of 18! + 1
ELIMINATE A

B. 17
Since 18! = (18)(17)(16)....(3)(2)(1), we can see that 17 is a factor (divisor) of 18!
So, according to Concept #1, 17 is NOT a divisor of 18! + 1
ELIMINATE B

C. 19
Leave for now.

D. 33
Since 18! = (18)(17)(16)..(11)..(3)(2)(1), we can see that 33 is a factor (divisor) of 18!
So, according to Concept #1, 33 is NOT a divisor of 18! + 1
ELIMINATE D

E. 39
Since 18! = (18)(17)(16)..(13)..(3)(2)(1), we can see that 39 is a factor (divisor) of 18!
So, according to Concept #1, 39 is NOT a divisor of 18! + 1
ELIMINATE E

By the process of elimination, the correct answer is C

Cheers,
Brent

Which of the following is a factor of 18!+1?
A. 15
B. 17
C. 19
D. 33
E. 39

c. 19

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