Find all pairs \((x,n)\) of positive integers such that \(x^n + 2^n + 1\) is a divisor of \(x^{n+1} + 2^{n+1} + 1\).
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 \(\phi(mn) = \phi(m)\phi(n)\).
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 \(d(n)+d(n+1)=5\).