Found problems: 573
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 $a$.
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.
Determine two last digits of number $Q = 2^{2017} + 2017^2$
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 $13n$.
The combination to open a safe is a five-digit number. different, randomly selected from $2$ to $9$. To open the box strong, you also need a key that is labeled with the number $410639104$, which is the sum of all combinations that do not open the box. What is the combination that opens the safe?
Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.
Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?
Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$.
[b](1)[/b] Find a $6$-digit positive integer $N$ such that
\[\frac{u(N)}{N} = \frac{23}{35}.\]
[b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that
\[\frac{u(N)}{N} = \frac{23}{35}.\]
For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?
Put $S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}$. The last digit of $S$ is
(A): $1$ (B): $3$ (C): $5$ (D): $7$ (E): None of the above.
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$.
Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.