How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

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?

Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

Let $\overline{abcd}$ be $4$ digit number, such that we can do transformations on it. If some two neighboring digits are different than $0$, then we can decrease both digits by $1$ (we can transform $9870$ to $8770$ or $9760$). If some two neighboring digits are different than $9$, then we can increase both digits by $1$ (we can transform $9870$ to $9980$ or $9881$). Can we transform number $1220$ to: $a)$ $2012$ $b)$ $2021$

Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?

Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]

Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$.

Initially, a positive integer $N$ is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) [i]A possible sequence of moves is given by $5 \to 20 \to 140 \to 041=41$.[/i] Determine for which values of $N$ it is possible to obtain $1$ after a finite number of such moves.

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 $N$.

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.