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 \(ab \equiv 1 \pmod p\).
a) Two students need to be chosen to participate in a mathematical Olympiad from a class of 30 students. In how many ways can this be done?
b) In how many ways can a team of three students in the same class be chosen?
How many ways can Susan choose 4 colours from 7 different ones?