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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