Found problems: 408
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?
All decimal digits of some natural number are $1,3,7$, and $9$. Prove that one can rearrange its digits so as to obtain a number divisible by $7$.
Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.
Prove that there exist two powers of $7$ whose difference is divisible by $2021$.
Let the number $ p $ be a prime divisor of the number $ 2 ^ {2 ^ k} + 1 $. Prove that $ p-1 $ is divisible by $ 2 ^ {k + 1} $.
Let non-constant polynomial $f(x)$ with real coefficients is given with the following property:
for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$
An integer $n\ge1$ is [i]good [/i] if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$.
How many good integers are $n\ge 1$?
Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.
A quadruple $(p, a, b, c)$ of positive integers is a[i] karaka quadruple[/i] if
$\bullet$ $p$ is an odd prime number
$\bullet$ $a, b$ and $c$ are distinct, and
$\bullet$ $ab + 1$, $bc + 1$ and $ca + 1$ are divisible by $p$.
(a) Prove that for every karaka quadruple $(p, a, b, c)$ we have $p + 2 \le\frac{a + b + c}{3}$.
(b) Determine all numbers $p$ for which a karaka quadruple $(p, a, b, c)$ exists with $p + 2 =\frac{a + b + c}{3}$
Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?
For which $m > 1$ is $(m -1)!$ divisible by $m$?
Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have:
$$\sum_{d | n} a_d = 2^n.$$
Show for every $n \in N$ that $n$ divides $a_n$.
Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.
Is there a number whose digits are only $1$'s and which is divided by $1999$?
Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares.
a) Give an example of such numbers $a, b$ and $c$.
b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$
Let $n$ be a positive integer and $p > n+1$ a prime.
Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$
Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.
Decide whether there is an integer $n > 1$ with the following properties:
(a) $n$ is not a prime number.
(b) For all integers $a$, $a^n - a$ is divisible by $n$
Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.
Find the smallest number of the form $1...1$ in its decimal expression which is divisible by $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.
Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.
Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$
Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$
Prove that among any $k + 1$ natural numbers there are two numbers whose difference is divisible by $k$.
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?