Found problems: 788
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?
The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?
The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?
A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.
Below the standard representation of a positive integer $n$ is the representation understood by $n$ in the decimal system, where the first digit is different from $0$. Everyone positive integer n is now assigned a number $f(n)$ by using the standard representation of $n$ last digit is placed before the first.
Examples: $f(1992) = 2199$, $f(2000) = 200$.
Determine the smallest positive integer $n$ for which $f(n) = 2n$ holds.
All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers.
a) Find all possibilities for the three numbers that are not included in this number.
b) Give an example of such a number.
Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that:
$\bullet$ it is a $4$-digit number whose first digit is $5$.
$\bullet$ it is a multiple of $9$;
$\bullet$ The larger number is more likely to be a password than a smaller number.
Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed:
Alice: ”I have no idea what the number is.”
Bob: ”I have no idea too.”
After that both of them knew which number they should try first. Identify this number
A positive integer is a [i]palindrome [/i] if it is written identically forwards and backwards. For example, $285582$ is a palindrome. A six digit number $ABCDEF$, where $A, B, C, D, E, F$ are digits, is called [i]cozy [/i] if $AB$ divides $CD$ and $CD$ divides $EF$. For example, $164896$ is cozy. Determine all cozy palindromes.
A real number is called [i]triplex[/i] if it has a decimal representation in which none of $0$ and $3$ different digit occurs. Prove that every positive real number is the sum of nine triplex numbers.
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer
Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.
For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself.
Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$.
Show that there is no number $n$ with $f (n) = 19091997$.
Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?
The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$.
a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots.
b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots.
(I. Voronovich)
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 $k$.
Find digits $x, y, z$ such that the equality
\[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\]
holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.
Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.
Find the smallest natural number $n$ which has the following properties:
a) Its decimal representation has a 6 as the last digit.
b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.
Given a finite sequence of zeros and ones, which has two properties:
a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap);
b) if you add any digit to the right of the sequence, then property (a) will no longer hold true.
Prove that the first four digits of our sequence coincide with the last four
Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!
To write all consecutive natural numbers from $1ab$ to $ab2$ inclusive, $1ab1$ digits have been used. Determine how many more digits are needed to write the natural numbers up to $aab$ inclusive. Give all chances. ($a$ and $b$ represent digits)
Does there exist a positive integer such that its last digit is nonzero and that it becomes exactly two times bigger when the order of its digits is reversed?