Found problems: 408
Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold:
$$pq | n^p + 2$$
$$n + 2 | n^p + q^p.$$
Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$.
( V Senderov)
Six dice are strung on a rigid wire so that the wire passes through two opposite faces of each die. Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table so that the six-digit number formed by their top faces is divisible by $7$. (The faces of a die are numbered from $1$ to $6$, the sum of the numbers on opposite faces is always equal to $7$.)
(G Galperin)
Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$?
If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.
Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.
Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.
Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.
We have $n > 2$ nonzero integers such that everyone of them is divisible by the sum of the other $n - 1$ numbers, Show that the sum of the $n$ numbers is precisely $0$.
Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation
$$ a^2 + b^2 = c^2,$$
then:
1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $,
2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $,
3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.
Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$
Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers.
Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.
Let $a$ and $b$ be distinct positive integers, bigger that $10^6$, such that $(a+b)^3$ is divisible with $ab$. Prove that $ \mid a-b \mid > 10^4$
On blackboard there are $10$ different positive integers which sum is equal to $62$. Prove that product of those numbers is divisible with $60$
Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.
Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.
Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.
a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$.
b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.
Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if
(a) $d = 10$;
(b) $d = 11$?
(V Senderov)
How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?
Prove that there exists an in nite sequence $<a_n>$ of positive integers such that for each $k \ge 1$
$(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties:
(i) $n$ has exactly $k$ digits (in decimal representation),
(ii) all the digits of $n$ are odd,
(iii) $n$ is divisible by $5$,
(iv) the number $m = n/5$ has $k$ odd digits
How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both?
How many pairs of integers $x, y$ are there between $1$ and $1000$ such that $x^2 + y^2$ is divisible by $7$?
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?
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.