A [i]complete [/i] number is a $9$ digit number that contains each of the digits $1$ to $9$ exactly once. The [i]difference [/i] number of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the [i]difference [/i] number of $25143$ is equal to $3431$. The [i]complete [/i] number $124356879$ has the additional property that its [i]difference [/i] number, $12121212$, consists of digits alternating between $1$ and $2$. Determine all $a$ with $3 \le a \le 9$ for which there exists a [i]complete [/i] number $N$ with the additional property that the digits of its [i]difference[/i] number alternate between $1 $ and $a$.

Let $k$ and $d$ be integers such that $k>1$ and $0\leq d<9$. Prove that there exists some integer $n$ such that the $k$th digit from the right of $2^n$ is $d$. [i](tatari/nightmare)[/i]

Let $a$ and $b$ be positive integers such that (i) both $a$ and $b$ have at least two digits; (ii) $a + b$ is divisible by $10$; (iii) $a$ can be changed into $b$ by changing its last digit. Prove that the hundreds digit of the product $ab$ is even.

Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary. [i]I. Dimovski[/i]

Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.

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 $k$.

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition: [i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\ Determine the smallest possible value of $N$.

Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?

Find $5$ last digits of the number $M = 5^{2010}$ . (A): $65625$, (B): $45625$, (C): $25625$, (D): $15625$, (E) None of the above.

The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

Ana and Banana are playing a game. Initially, Ana secretly picks a number $1\leq A\leq 10^6$. In each subsequent turn of the game, Banana may pick a positive integer $B$, and Ana will reveal to him the most common digit in the product $A\cdot B$ (written in decimal notation). In the case when at least two digits are tied for being the most common, Ana will reveal all of them to Banana. For example, if $A\cdot B=2022$, Ana will tell Banana that the digit $2$ is the most common, while if $A\cdot B=5783783$, Ana will reveal that $3, 7$ and $8$ are the most common. Banana's goal is to determine with certainty the number $A$ after some number of turns. Does he have a winning strategy?

Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.

Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number. Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.

Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.

Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?