Found problems: 788
Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.
The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.
Let $f(k)$ be the number of integers $n$ satisfying the following conditions:
(i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed;
(ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$.
Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$.
[i]Proposed by Dirk Laurie, South Africa[/i]
Determine the number of three-digit numbers with the following property:
The number formed by the first two digits is prime and the number formed by the last two digits is prime.
Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$
The decimal representation of $$\dfrac{1}{20^{20}}$$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
$\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$
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)
For each positive integer $n$, let $c(n)$ be the number of digits of $n$. Let $A$ be a set of positive integers with the following property: If $a$ and $b$ are two distinct elements in $A$, then $c(a +b)+2 > c(a)+c(b)$. Find the largest number of elements that $A$ can have.
PS. In the original wording: c(n) = ''cantidad de dıgitos''
Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria:
1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$,
2. $N$ only has three different digits, and none of them are zero,
3. Exactly 99.9% of the digits of $N$ are odd.
Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$
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?
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.
The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$.
(a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system.
(b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.
Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.
Let $X= a1b9$ and $Y ab = 51ab$ be two positive integers where $a$ and $b$ are digits. $X$ is known to be multiple of a positive two-digit number $n$ and $Y$ is the next multiple of that number $n$. Find the number $n$ and the digits $a$ and $b$. Justify why there are no other possibilities.
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 $22$?
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)
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 $m$?
What is the smallest positive multiple of $225$ that can be written using
digits $0$ and $1$ only?
Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$.
(A Galochkin)
On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.
Find, with proof, all positive integers $n$ such that neither $n$ nor $n^2$ contain a $1$ when written in base $3$.