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.
The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than 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.
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.
Determine all natural numbers \(m\) and \(n\) such as \(m! + 12 = n^2\).
Is it possible to:
a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?
b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?
Solve the equation \((x + 1)^3 = x^3\).