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Find the largest number of which each digit, starting with the third, is equal to the sum of the two previous digits.

Find the largest six-digit number, for which each digit, starting with the third, is equal to the sum of the two previous digits.

In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

One three-digit number consists of different digits that are in ascending order, and in its name all words begin with the same letter. The other three-digit number, on the contrary, consists of identical digits, but in its name all words begin with different letters. What are these numbers?

A girl chose a 4-letter word and replaced each letter with the corresponding number in the alphabet. The number turned out to be 2091425. What word did she choose?

Try to find all natural numbers which are five times greater than their last digit.

A three-digit number $ABB$ is given, the product of the digits of which is a two-digit number $AC$ and the product of the digits of this number is $C$ $($here, as in mathematical puzzles, the digits in the numbers are replaced by letters where the same letters correspond to the same digits and different letters to different digits$)$. Determine the original number.

The best student in the class, Katie, made up a huge number, writing out in a row all of the natural numbers from 1 to 500: 123 … 10111213 … 499500. The second-best student, Tom, erased the first 500 digits of this number. What do you think, what number does the remaining number begin with?

The best student in the class, Katie, and the second-best, Mike, tried to find the minimum 5-digit number which consists of different even numbers. Katie found her number correctly, but Mike was mistaken. However, it turned out that the difference between Katie and Mike’s numbers was less than 100. What are Katie and Mike’s numbers?

7 different digits are given. Prove that for any natural number n there is a pair of these digits, the sum of which ends in the same digit as the number.

What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?

Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.

Two people play a game with the following rules: one of them guesses a set of integers $(x_1, x_2, …, x_n)$ which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum $a_1x_1 + … + a_nx_n$, where $(a_1, … ,a_n)$ is any set. What is the smallest number of questions for which the guesser recognizes the intended set?

Note that if you turn over a sheet on which numbers are written, then the digits 0, 1, 8 will not change and the digits 6 and 9 will switch places, whilst the others will lose their meaning. How many nine-digit numbers exist that do not change when a sheet is turned over?

An infinite sequence of digits is given. Prove that for any natural number $n$ that is relatively prime with a number 10, you can choose a group of consecutive digits, which when written as a sequence of digits, gives a resulting number written by these digits which is divisible by $n.$

Prove that for any positive integer n, it is always possible to find a number, consisting of digits 1 and 2, that is divisible by $2^n$.

$($For example, 2 divides by 2, 12 divides by 4, 112 divides by 8, 2112 divides by 16 and so on …$)$.

To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.

Without calculating the answer to $2^{30}$, prove that it contains at least two identical digits.