For-each-positive-integer-n-consider-the-highest-common-factor-h-n-of-the-two-numbers-n-1-and-n-1-For-n-lt-100-find-the-largest-value-of-h-n-

Question Number 21230 by Tinkutara last updated on 16/Sep/17

For each positive integer n, consider

the highest common factor h_{n} of the two

numbers n! + 1 and (n + 1)! . For n < 100,

find the largest value of h_{n} .

Answered by dioph last updated on 17/Sep/17

as the numbers from 2 until n do not

divide n! + 1, h_{n} is either 1 or n + 1 .

h_{n} = n + 1 \Leftrightarrow n! + 1 = k (n + 1)

\Leftrightarrow n! \equiv - 1 (\mod n + 1)

using Wilsons theorem, this happens

only when n + 1 is prime .

The last prime before 100 is 97 so

for 96! + 1 and 97! we have h_{n} = 97

Commented by Tinkutara last updated on 17/Sep/17

How do you used {Wilson}^{'} s theorem ?