Found problems: 788
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?
Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits?
[i]Proposed by A. Shapovalov[/i]
Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit
number with all digits equal.
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)
[i](Karl Czakler)[/i]
Find a number that increases by a factor of $1996$ if the digits in the first and fifth places after the decimal place are swapped in its decimal notation.
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.
Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.
A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits.
Find all pairs of consecutive numbers such that both are tricubic.
On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?
The natural numbers are written in succession, forming a sequence of digits$$12345678910111213141516171819202122232425262728293031\ldots$$Determine how many digits the natural number has that contributes to this sequence with the digit in position $10^{2000}$.
Clarification: The natural number that contributes to the sequence with the digit in position $10$ has $2$ digits, because it is $10$; The natural number that contributes to the sequence with the digit at position $10^2$ has $2$ digits, because it is $55$.
Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.
Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.
We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true :
$\bullet$ $a\ge b$
$\bullet$ $ab - cd = cd -ba$.
For example, $2011$ is pora because $20-11 = 11-02$
Find all the numbers around.
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.
The number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$ is
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
A schoolgirl forgot to write a multiplication sign between two $3$-digit numbers and wrote them as one number. This $6$-digit result proved to be $3$ times greater than the product (obtained by multiplication). Find these numbers.
(A Kovaldzhi,
Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that
\[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?
Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.
Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements:
Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors".
Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation".
Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer"
Stefan:"Everyone of them has exactly one prime divisor less than $10$".
Filip:"The three numbers are square free".
Professor found the right answer. Which numbers did he mention?
For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times.
Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.
How many positive five-digit integers are there that have the product of their five digits equal to $900$?
(Karl Czakler)