Found problems: 788
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.
Find all the natural numbers that are $300$ times the sum of its digits.
Let S(n) be the sum of the digits of the positive integer $n$. Determine
$$S(S(S(2003^{2003}))).$$
Find all 2-digit numbers$ n$ having the property:
'Number $n^2$ is 4-digit number of form $\overline{xxyy}$.
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.
Prove that there is no natural number n such that the sum of all the digits of the number m, where $m = n(2n-1)$ is equal to $2000$.
Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$
Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?
Determine the largest two-digit number $d$ with the following property:
for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.
Note The numbers $a \ne 0, b$ and $c$ need not be different.
Determine whether there exists a positive integer $n$ such that the sum of the digits of $n^2$ is $2002$.
The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.
We increased some positive integer by $10\%$ and obtained a positive integer. Is it possible that in doing so we decreased the sum of digits exactly by $10\%$ ?
Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$.
[hide=original wording]
Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]
Define [i]glueing[/i] of positive integers as writing their base ten representations one after another and
interpreting the result as the base ten representation of a single positive integer.
Find all positive integers $k$ for which there exists an integer $N_k$ with the following property: for all $n \ge N_k$, we can glue the numbers $1,2,\dots,n$ in some order so that the result is a number divisible by $k$.
[i]Remark[/i]. The base ten representation of a positive integer never starts with zero.
[i]Example[/i]. Glueing $15, 14, 7$ in this order makes $15147$.
The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times .
(a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits?
(b) Prove that there must be a row or column containing more than $3$ different digits .
{ L . D . Kurlyandchik , Leningrad)
Consider the natural number prime $p, p> 5$. From the decimal number $\frac1p$, randomly remove $2012$ numbers, after the comma. Show that the remaining number can be represented as $\frac{a}{b}$ , where $a$ and $b$ are coprime numbers , and $b$ is multiple of $p$.
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 $s(n) > s(2n)$.
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 $\frac{b}{a}= 6$.
The positive integer $ n$ has the following property:
if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains.
Find $n$.
Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.
a) Prove that the fraction $\frac{3n+5}{2n+3}$ is irreducible for every $n \in N$
b) Let $x,y$ be digits of decimal representation system with $x>0$, and $\frac{\overline{xy}+12}{\overline{xy}-3}\in N$, prove that $x+y=9$. Is the converse true?
(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares.
(b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.
Juan writes the list of pairs $(n, 3^n)$, with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?
Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.
Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$.