Problems

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 \(a\).

Given a pile of five cards, is it true that reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

Today we will discover some ideas related to non-isosceles triangles, this particular restriction comes from the fact that in isosceles triangles a median and a height coincide.

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 \(n\).

With a pile of four cards, does reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

You have in your hands a royal flush! That is, Ace, King, Queen, Jack and \(10\) of spades. How many shuffles of your hand swap the Ace and Jack?

In the diagram below, I wish to write the numbers \(6, 11, 19, 23, 25, 27\) and \(29\) in the squares, but I want the sum of the numbers in the horizontal row to equal the sum of the numbers in the vertical column. What number should I put in the blue square with the question mark?