prove-that-if-n-N-n-gt-1-and-n-is-odd-then-1-n-n-1-n-is-divisible-by-n-dont-use-modn-

Question Number 192220 by gatocomcirrose last updated on 12/May/23

prove that if n \in N, n > 1 and n is odd then

1^{n} + \dots + {(n - 1)}^{n} is divisible by n

(dont use \equiv (modn))

Commented by AST last updated on 12/May/23

I t f o l l o w s f r o m t h e f a c t t h a t a^{n} + b^{n} i s d i v i s i b l e

b y a + b w h e n n i s o d d .

{1^{n} + {(n - 1)}^{n}} + {2^{n} + {(n - 2)}^{n}} + \dots + {{(\frac{n - 1}{2})}^{n} + {(\frac{n + 1}{2})}^{n}}

Commented by gatocomcirrose last updated on 12/May/23

ohhhh yeahh thanks bro

Answered by Frix last updated on 12/May/23

1^{n} + {(n - 1)}^{n} = 1^{n} + n \times (\dots) - 1^{n} = n \times (\dots)

2^{n} + {(n - 2)}^{n} = 2^{n} + n \times (\dots) - 2^{n} = n \times (\dots)

\dots

You get tbe idea ?

Commented by gatocomcirrose last updated on 12/May/23

yeah thank you!