Found problems: 573
Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.
For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime?
[i]Proposed by Oleksii Masalitin[/i]
We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits?
(Paolo Leonetti)
Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?
Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.
In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions:
a) Can there be only two terms with a certain number of digits?
b) Can there be five consecutive terms with the same number of digits?
c) Can there be four terms of n digits, followed by four with $n + 1$ digits?
d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?
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.
How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?
All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.
We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met:
[list]
[*]$M$ is a perfect square.
[*]The number formed by the first $n$ digits of $M$ is a perfect square.
[*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero).
[/list]
Find a hypersquared number with $2000$ digits.
A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.
A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits.
For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$.
Determine all common sequences.
[i]Proposed by Wong Jer Ren[/i]
Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.
Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.
The last digit of number $2017^{2017} - 2013^{2015}$ is
(A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.
Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.
Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$.
[list=a]
[*] Determine its units digit.
[*] Determine its tens digit.
[/list]
Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.