Rational numbers \(x,y,z\) are such that all the numbers \(x+y^2+z^2\), \(x^2+y+z^2\), \(x^2+y^2+z\) are integers. Prove that \(2x\) is also an integer.
Prove that the product of five consecutive integers is divisible by \(120\).
Show that there are infinitely many composite numbers \(n\) such that \(3^{n-1}-2^{n-1}\) is divisible by \(n\).
Show that there are infinitely many numbers \(n\) such that \(2^n+1\) is divisible by \(n\). Find all prime numbers, that satisfy this property.
If \(k>1\), show that \(k\) does not divide \(2^{k-1}+1\). Find all prime numbers \(p,q\) such that \(2^p+2^q\) is divisible by \(pq\).
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 \(p\).