Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?

How many zeros does $101^{100} - 1$ end with?

Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.

Find all $6$ digit natural numbers, which consist of only the digits $1,2,$ and $3$, in which $3$ occurs exactly twice and the number is divisible by $9$.

Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.

$72$ was added to the natural number $n$ and in the sum we got a number written in the same digits as the number $n$, but in the reverse order. Find all numbers $n$ that satisfy the given condition.

A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.

The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?

Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

Let $N$ be a positive integer. Initially, a positive integer $A$ is written on the board. At each step, we can perform one of the following two operations with the number written on the board: (i) Add $N$ to the number written on the board and replace that number with the sum obtained; (ii) If the number on the board is greater than $1$ and has at least one digit $1$, then we can remove the digit $1$ from that number, and replace the number initially written with this one (with removal of possible leading zeros) For example, if $N = 63$ and $A = 25$, we can do the following sequence of operations: $$25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5$$ And if $N = 143$ and $A = 2$, we can do the following sequence of operations: $$2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3$$ For what values of $N$ is it always possible, regardless of the initial value of $A$ on the blackboard, to obtain the number $1$ on the blackboard, through a finite number of operations?

Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?

How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.

Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation. For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.

Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?

The teacher writes a $2002$-digit number consisting only of digits $9$ on the blackboard. The first student factors this number as $ab$ with $a > 1$ and $b > 1$ and replaces it on the blackboard by two numbers $a'$ and $b'$ with $|a-a'| = |b-b'| = 2$. The second student chooses one of the numbers on the blackboard, factors it as $cd$ with $c > 1$ and $d > 1$ and replaces the chosen number by two numbers $c'$ and $d'$ with $|c-c'| = |d-d'| = 2$, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to $9$?

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.