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 integer solutions of the equation \(3x - 12y = 7\).
Determine all the integer solutions for the equation \(21x + 48y = 6\).
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.
Let \(s\) and \(t\) be two positive integers. Can we have \(s^2=t^2+2\)?