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Which of the following is a factor of 18!+1?

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Which of the following is a factor of 18!+1? [#permalink] New post 08 May 2016, 02:51
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Which of the following is not a factor of 18!+1?
A. 15
B. 17
C. 19
D. 33
E. 39

Kudos for best explanation! This is a tricky one!
[Reveal] Spoiler: OA

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Re: Which of the following is a factor of 18!+1? [#permalink] New post 09 May 2016, 15:11
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soumya1989 wrote:
Which of the following is a factor of 18!+1?
A. 15
B. 17
C. 19
D. 33
E. 39

Kudos for best explanation! This is a tricky one!



In solving this question, we must remember the rule that two consecutive integers will never share the same prime factors. Thus, since 18! and 18! + 1 are two consecutive integers, they do not share the same prime factors.

Also we must remember that n! is divisible by any prime number less than or equal to n, but it’s not divisible by any prime number greater than n. For example, 5! is divisible by 2, 3 and 5, but it’s not divisible by 7, 11, 13, etc. Furthermore, n! is divisible by the product of any two distinct prime numbers less than or equal to n. For example, 5! is divisible by 6 (which 2 x 3), 10 (which is 2 x 5) and 15 (which is 3 x 5).

Therefore, 18! is divisible by 15 (which is 3 x 5), 17, 33 (which is 3 x 11), and 39 (which is 3 x 13). However 18! is not divisible by 19 since 19 > 18. Since 18! and 18! + 1 do not share the same prime factors, 19 must be a prime factor of 18! + 1.

Answer: C
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Re: Which of the following is a factor of 18!+1? [#permalink] New post 18 Dec 2016, 13:55
18! is divisible by 15 (which is 3 x 5), 17, 33 (which is 3 x 11), and 39 (which is 3 x 13). How can you write this, we know that n! is divisible by any prime number less than or equal to n, then 18! is divisible by any prime number less than or equal to n, we found that 18! is divisible by only 2, 3 and 3, i cant understand your explanation! please, can you explain it again? thanks!
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Re: Which of the following is a factor of 18!+1? [#permalink] New post 18 Dec 2016, 14:11
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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.
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Re: Which of the following is a factor of 18!+1? [#permalink] New post 09 Aug 2018, 06:59
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.
Re: Which of the following is a factor of 18!+1?   [#permalink] 09 Aug 2018, 06:59
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