Numbers \(a, b, c\) are integers with \(a\) and \(b\) being coprime. Let us assume that integers \(x_0\) and \(y_0\) are a solution for the equation \(ax + by = c\).
Prove that every solution for this equation has the same form \(x = x_0 + kb\), \(y = y_0 - ka\), with \(k\) being a random integer.
There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.
Prove the mass of all the weights is the same, if it is known that:
a) the mass of each weight in grams is an integer;
b) the mass of each weight in grams is a rational number;
c) the mass of each weight could be any real (not negative) number.
A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be exactly seven times their product. Determine these numbers.
The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.
Solve the equation \(3x + 5y = 7\) in integers. Make sure that you’ve found all integer solutions.
Determine all the integer solutions for the equation \(21x + 48y = 6\).
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
How many rational terms are contained in the expansion of
a) \((\sqrt 2 + \sqrt[4]{3})^{100}\);
b) \((\sqrt 2 + \sqrt[3]{3})^{300}\)?
Determine all natural numbers \(m\) and \(n\) such as \(m! + 12 = n^2\).