Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.
Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).
Let \(m\) and \(n\) be integers. Prove that \(mn(m + n)\) is an even number.
Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.
Prove that any \(n\) numbers \(x_1,\dots , x_n\) that are not pairwise congruent modulo \(n\), represent a complete system of residues, modulo \(n\).
Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.
Without calculating the answer to \(2^{30}\), prove that it contains at least two identical digits.
For what natural numbers \(a\) and \(b\) is the number \(\log_{a} b\) rational?
Prove that if \((m, 10) = 1\), then there is a repeated unit \(E_n\) that is divisible by \(m\). Will there be infinitely many repeated units?
Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.