Solve the equation \(3x + 5y = 7\) in integers.
Determine all integer solutions of the equation \(3x - 12y = 7\).
Prove there are no integer solutions for the equation \(4^k - 4^l = 10^n\).
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.
Solve the equation \(x + \frac{1}{(y + 1/z)}= 10/7\) in natural numbers.