Found problems: 159
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.
For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.
Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.
Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$,
that is, $n = 1234567891011...40214022$.
a. Show that $n$ is divisible by $3$.
b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$
Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$.
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?
(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$?
b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?
For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern:
$A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$
$T(n_1)=T(n_2)=...=T(n_{148})$
$T(m_1)=T(m_2)=...=T(m_{148})$
Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).
Find all positive integers that are equal to $700$ times the sum of its digits.
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits?
[i]Proposed by A. Shapovalov[/i]
Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.
For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.
The number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$ is
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd.
(D Fomin, St Petersburg)
Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?
We consider the sum of the digits of a positive integer.
For example, the sum of the digits of $2007$ is equal to $9$, since $2 + 0 + 0 + 7 = 9$.
(a) How many integers $n$, where $0 < n < 100 000$, have an even sum of digits?
(b) How many integers $n$, where $0 < n < 100 000$, have a sum of digits that is less than or equal to $22$?
a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$.
b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.
Find all the natural numbers that are $300$ times the sum of its digits.
Let S(n) be the sum of the digits of the positive integer $n$. Determine
$$S(S(S(2003^{2003}))).$$
Prove that there is no natural number n such that the sum of all the digits of the number m, where $m = n(2n-1)$ is equal to $2000$.