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 \(d = \mathrm{GCD} (a, b)\).
Prove that for a real positive \(\alpha\) and a positive integer \(d\), \(\lfloor \alpha / d\rfloor = \lfloor \lfloor \alpha\rfloor / d\rfloor\) is always satisfied.
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 if \(p\) is a prime number and \(1 \leq k \leq p - 1\), then \(\binom{p}{k}\) is divisible by \(p\).
Prove that if \(p\) is a prime number, then \((a + b)^p - a^p - b^p\) is divisible by \(p\) for any integers \(a\) and \(b\).
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?