There are two piles of sweets: one with 20 sweets and the other with 21 sweets. In one go, one of the piles needs to be eaten, and the second pile is divided into two not necessarily equal piles. The player that cannot make a move loses. Which player wins and which one loses?
The game begins with the number 0. In one go, it is allowed to add to the actual number any natural number from 1 to 9. The winner is the one who gets the number 100.
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.
Solve the equation \(3x + 5y = 7\) in integers.
Determine all integer solutions of the equation \(3x - 12y = 7\).
Determine all the integer solutions for the equation \(21x + 48y = 6\).
A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by \(m\) squares to the right or by \(n\) squares to the left. For which \(m\) and \(n\) can it move to the next cell to the right?
Solve the equations \(x^2 = 14 + y^2\) in integers.
Solve the equation with integers \(x^2 + y^2 = 4z - 1\).
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).