Four numbers (from 1 to 9) have been used to create two numbers with four-digits each. These two numbers are the maximum and minimum numbers, respectively, possible. The sum of these two numbers is equal to 11990. What could the two numbers be?
Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number than is divisible by 1987.
Prove that there is a power of 3 that ends in 001.
How many six-digit numbers exist, the numbers of which are either all odd or all even?
Prove that the product of any three consecutive natural numbers is divisible by 6.
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?
Prove that for any number \(d\), which is not divisible by \(2\) or by \(5\), there is a number whose decimal notation contains only ones and which is divisible by \(d\).