Found problems: 573
Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.
Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number.
Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.
A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences.
If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.
A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?
We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
Given the following list of numbers:
$$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$
where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?
How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?
Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$?
$\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.
A set of five-digit numbers $\{N_1, ...,N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1$, $...$ , $N_k$. Find the smallest possible value of $k$.
Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.
Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.
The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is
(A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.
Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.
Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$
Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$
The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?
Determine the sum of all positive integers $n$ that satisfy the following condition:
when $6n + 1$ is written in base $10$, all its digits are equal.