Notice that the square number 1089 \((=33^2)\) has two even and two odd digits in its decimal representation.
(a) Can you find a 6-digit square number with the same property (the number of odd digits equals the number of even digits)?
(b) What about such 100-digit square number?
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
Prove the magic trick for the number \(1089 = 33^2\): if you take any \(3\)-digit number \(\overline{abc}\) with digits coming in strictly descending order and subtract from it the number obtained by reversing the digits of the original number \(\overline{abc} - \overline{cba}\) you get another \(3\)-digit number, call it \(\overline{xyz}\). Then, no matter which number you started with, the sum \(\overline{xyz} + \overline{zyx} = 1089\).
Recall that a number \(\overline{abc}\) is divisible by \(11\) if and only if \(a-b+c\) also is.
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
A monkey becomes happy when they eat three different fruits. What is the largest number of monkeys that can become happy with \(20\) pears, \(30\) bananas, \(40\) peaches and \(50\) tangerines?
Given a natural number \(n\), find a formula for the number of \(k\) less than \(n\) such that \(k\) is coprime to \(n\). Prove that the formula works.