Find all positive integers \((x,n)\) such that \(x^n + 2^n + 1\) is a divisor \(x^{n+1} + 2^{n+1} + 1\).
Show that if \(n\) is an integer, greater than \(1\), then \(n\) does not divide \(2^n-1\).
Find all the integers \(n\) such that \(1^n + 2^n + ... + (n-1)^n\) is divisible by \(n\).
Find out how many are there integers \(n>1\) such that the number \(a^{25}-a\) is divisible by \(n\) for any 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 number. Denote by \(\phi(n)\) the number of integers from \(1\) to \(n-1\) coprime with \(n\). Find \(\phi(n)\) for 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 number, \(a\) be an integer, coprime with \(n\). Prove that \(a^{\phi(n)-1}-1\) is divisible by \(n\).
Let \(\phi(n)\) be the Euler’s function, namely the amount of numbers from \(1\) to \(n\), coprime with \(n\). For two natural numbers \(m,n\) such that \(\mathbb{GCD}(m,n)=1\) prove that \(\phi(mn) = \phi(m)\phi(n)\).
For any positive integer \(k\), the factorial \(k!\) is defined as a product of all integers between 1 and \(k\) inclusive: \(k! = k \times (k-1) \times ... \times 1\). What’s the remainder when \(2025!+2024!+2023!+...+3!+2!+1!\) is divided by \(8\)?
Let \(n\) be a natural number, and let \(d(n)\) be the number of factors of \(n\). For example, the factors of \(6\) are \(1,2,3,6\), so \(d(6)=4\). Find all \(n\) such that \(d(n)+d(n+1)=5\).