Found problems: 159
A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$.
a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers.
b) Does there exist a positive integer such that each of its multiples is typical?
The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a]
[*] the square of this number;
[*]the fourth power of this number,
[/list] given that $k\geqslant 4.$
[i]From the folklore[/i]
Determine all natural numbers $m$ and $n$ such that
\[
n \cdot (n + 1) = 3^m + s(n) + 1182,
\]
where $s(n)$ represents the sum of the digits of the natural number $n$.
Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$.
a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \le k \le 1992, $ but $T(1993 \cdot N)$ is odd.
b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.
Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$
[hide= original wording] before finding a typo ..
Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $1,0,0,4.$
Posts 2 and 3 reply to this wording
[/hide]
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$?
(AI Galochkin)
Prove that among any $18$ consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.
A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which
\[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\]
where $s(n)$ denotes the sum of digits of $n$ in decimal representation.
Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.