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 integers \(n\) 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 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 \(p\).