Found problems: 788
Find a positive integer $x$ such that the sum of the digits of $x$ is greater than $2011$ times the sum of the digits of the number $3x$ ($3$ times $x$).
Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$.
Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.
Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.
Find all positive integers $\overline{xyz}$ ($x$, $y$ and $z$ are digits) such that
$\overline{xyz} = x+y+z+xy+yz+zx+xyz$
Let $a$ and $b$ be positive integers, $a < b$, such that in the decimal expansion of the fraction $\dfrac{a}{b} $ the five digits $1,4,2,8,6$ appear somewhere, in that order and consecutively. Determine the lowest possible value $b$ can take .
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 $2$?
For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number.
Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.
Find unknown digits $a$ and $b$ such that number $\overline{a783b}$ is divisible with $56$
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.
Let's call a ticket with a number from $000000$ to $999999$ [i]excellent [/i] if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.
A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three ($6$ digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by $13$.
Every officially published book used to have an ISBN code (International Standard Book Number) which consisted of $10$ symbols. Such code looked like this: $$a_1a_2 . . . a_9a_{10}$$ with $a_1, . . . , a_9 \in \{0, 1, . . . , 9\}$ and $a_{10} \in \{0, 1, . . . , 9, X\}$. The symbol $X$ stood for the number $10$. With a valid ISBN code was
$$a_1 + 2a2 + . . . + 9a_9 + 10a_{10}$$ a multiple of $11$. Prove the following statements.
(a) If one symbol is changed in a valid ISBN code, the result is no valid ISBN code.
(b) When two different symbols swap places in a valid ISBN code then the result is not a valid ISBN.
Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.
Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.
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 $2$.
In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded)
\[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \]
as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.
It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.
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.
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]
Find the last two digits of $11^{1998}$
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
Determine all ten-digit numbers whose decimal $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation.
That is: the first digit is equal to the amount of "$0$" in the writing of that number, the second digit is equal to the amount of "$1$" in the writing of that number, the third digit is equal to the amount of "$2$" in the writing of that number, ... , the tenth digit is equal to the number of "$9$" in the writing of that number.
Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)
Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its digits?