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 \(n\).
Show that a rotation about an axis on a sphere followed by a rotation about a different axis is again a rotation about some axis.
Let \(u\) and \(v\) be two positive integers, with \(u>v\). Prove that a triangle with side lengths \(u^2-v^2\), \(2uv\) and \(u^2+v^2\) is right-angled.
We call a triple of natural numbers (also known as positive integers) \((a,b,c)\) satisfying \(a^2+b^2=c^2\) a Pythagorean triple. If, further, \(a\), \(b\) and \(c\) are relatively prime, then we say that \((a,b,c)\) is a primitive Pythagorean triple.
Show that every primitive Pythagorean triple can be written in the form \((u^2-v^2,2uv,u^2+v^2)\) for some coprime positive integers \(u>v\).
The lengths of three sides of a right-angled triangle are all integers.
Show that one of them is divisible by \(5\).
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 \(n\).