Found problems: 788
Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.
Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.
Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.
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?
An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation.
(a) Prove that the number of its terms is less than $100$.
(b) Give an example of such a progression with $72$ terms.
(c) Prove that the number of terms in any such progression does not exceed $72$.
(V. Bugaenko and Tarasov, Moscow)
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
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 $a$.
Find all the $4$-digit natural numbers, written in base $10$, that are equal to the cube of the sum of its digits.
Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.
The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$.
(a) Find one such number.
(b) Find the smallest such number.
Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.
(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$.
(b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.
Show that there exists a real constant $C>1$ with the following property:
For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.)
[i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]
Find all natural numbers $N$ (in decimal system) with the following properties:
(i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes,
(ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.
Find the last two digits of each of the numbers $3^{1974}$ and $7^{1974}$.
Peter added a positive integer $M{}$ to a positive integer $N{}$ and noticed that the sum of the digits of the resulting integer is the same as the sum of the digits of $N{}$. Then he added $M{}$ to the result again, and so on. Will Peter eventually get a number with the same digit sum as the number $N{}$ again?
Determine two last digits of number $Q = 2^{2017} + 2017^2$
We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?
Roma wrote on the board each of the numbers $2018, 2019, 2020$, $100$ times each. Let us denote by $S(n)$ the sum of digits of positive integer $n$. In one action, Roma can choose any positive integer $k$ and instead of any three numbers $a, b, c$ written on the board write the numbers $2S(a + b) + k, 2S(b + c) + k$ and $2S(c + a) + k$. Can Roma after several such actions make $299$ numbers on the board equal, and the last one differing from them by $1$?
[i]Proposed by Oleksii Masalitin[/i]
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Find all polynomials $P$ with integer coefficients such that $$s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).$$
for all $x,y\in \mathbb{N}$.
Note: $s(x)$ denotes the sum of digits of $x$.
[i]Proposed by Şevket Onur YILMAZ[/i]
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
Let $A$ be a six-digit number, three of which are colored and equal to $1, 2$, and $4$.
Prove that it is always possible to obtain a number that is a multiple of $7$, by performing only one of the following operations: either delete the three colored figures, or write all the numbers of $A$ in some order.
Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.