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 \(d\).
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.
To transmit messages by telegraph, each letter of the Russian alphabet () ( and are counted as identical) is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet (letter numbering starts from zero). For example, the letter is represented in the form 00000, letter -00001, letter -10111, letter -11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters is received. Find the word you sent.
How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.
Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.
How many six-digit numbers exist, for which each succeeding number is smaller than the previous one?