A player in the card game Preferans has 4 trumps, and the other 4 are in the hands of his two opponents. What is the probability that the trump cards are distributed a) \(2: 2\); b) \(3: 1\); c) \(4: 0\)?
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.
Could it be that a) \(\sigma(n) > 3n\); b) \(\sigma(n) > 100n\)?
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.
The numbers \(1, 2,\dots ,99\) are written on 99 cards. Then the cards are shuffled and placed with the number facing down. On the blank side of the cards, the numbers \(1, 2, \dots , 99\) are once again written.
The sum of the two numbers on each card are calculated, and the product of these 99 summations is worked out. Prove that the end result will be an even number.
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.