Let \(n>1\) be an integer. Show that \(n\) does not divide \(2^n-1\).
Find all integers \(n\) such that \(1^n + 2^n + ... + (n-1)^n\) is divisible by \(n\).
How many integers are there \(n>1\) such that \(a^{25}-a\) is divisible by \(n\) for every integer \(a\).
Let \(p\) be a prime number, \(a\) be an integer, not divisible by \(p\). Prove that \(a^p-a\) is divisible by \(p\).
Let \(n\) be an integer. Denote by \(\phi(n)\) the number of integers from \(1\) to \(n-1\) coprime to \(n\). Find \(\phi(n)\) in the following cases:
\(n\) is a prime number.
\(n = p^k\) for a prime \(p\).
\(n=pq\) for two different primes \(p\) and \(q\).
Let \(n\) be an integer and let \(a\) be an integer coprime to \(n\). Prove that \(a^{\phi(n)-1}-1\) is divisible by \(n\).
Let \(\phi(n)\) be Euler’s function. Namely \(\phi(n)\) counts how many integers from \(1\) to \(n\) inclusive are coprime with \(n\). For two natural numbers \(m\), \(n\) such that \(\gcd(m,n)=1\), prove that \(\phi(mn) = \phi(m)\phi(n)\).
Let \(a\) and \(b\) be two different \(9\)-digit numbers. It is known that each one of them contains all of the digits \(1,2,...9\). Find the maximal value of \(\gcd(a,b)\).
For an odd number \(N\) denote by \(A\) the minimal positive difference between prime divisors of \(N\), denote by \(B\) the minimal positive difference between composite divisors of \(N\). Usually we have \(A<B\), but can we have \(A>B\)? (Disregard numbers such as \(15\) where one of \(A\) or \(B\) is not defined)
A natural number \(N\) is called perfect if it equals the sum of its divisors, except for \(N\) itself. Prove that if \(2^r-1\) is prime, then \((2^r-1)2^{r-1}\) is a perfect number. Are there any odd perfect numbers?