Number Theory

Number Theory Level 2

Is there any even positive integer n ≠ 2 n \ne 2 , such that ( n − 1 ) ! (n-1)! is not divisible by n n ?

Notation: ! ! denotes the factorial notation .

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Arunsoumya Basu by Arunsoumya Basu , 18, India

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2 solutions

Anthony Holm
Jun 19, 2017

Any prime number, p, will work. Because no natural number less than p has a prime factor of p, obviously p cannot be a prime factor of (p-1)! as (p-1)! has only the prime factors of every number less than p.

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Yes, but the question is, is there any even positive integer n except n=2, such that the condition is true ?

Arunsoumya Basu - 3 years, 11 months ago

Only 1 example is enough: take n=4, then you get 3 ! = 6 3! = 6 . Which is not divisible by 4.

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