Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of \(d\). Prove that \(d >30,000\).
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 \(d = \mathrm{GCD} (a, b)\).
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.
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.
Solve the equations in integers:
a) \(3x^2 + 5y^2 = 345\);
b) \(1 + x + x^2 + x^3 = 2^y\).
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 \(n\).