Found problems: 387
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 $3$.
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 $.
Find the smallest positive integer $N$ satisfies :
1 . $209$│$N$
2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )
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 $a^n + b^n$.
Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.
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}] $.
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds
$$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$
We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.
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)
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 $a_1a_2a_3 ...a_k + 1$.
Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?
Show that the expression $$\frac{n^5 -5n^3 + 4n}{n + 2}$$ where n is any integer, it is always divisible by $24$.
1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ .
2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$
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
Prove that $3^n + 1$ is not divisible by $2^n$ for any integer $n > 1$.
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $.
Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and
$$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$
Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$.
Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.
Let $p$ be an odd prime number.
a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n.
b) Find all $n$ satisfy condition above when $p = 3$
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.
A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.