Found problems: 408
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.
Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$.
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Let $a, b, c$, and $d$ be four distinct integers.
Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$.
[i](Walther Janous)[/i]
For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors.
(M Levin)
A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$
Prove that the sum of the four prime numbers is divisible by $60$.
(Walther Janous)
Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.
Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the fi rst $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?
The sum of all even positive integers less than $100$ those are not divisible by $3$ is
(A): $938$, (B): $940$, (C): $1634$, (D): $1638$, (E): None of the above.
Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.
For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$.
Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$
Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$.
Malik Talbi
The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$, $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.
If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we defi ne $f_s(n) = d_1^s+d_2^s+..+d_k^s$.
For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$.
Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.
Let $p$ be an odd prime number. Prove that there exists a unique integer $k$ such that $0 \le k \le p^2$ and $p^2$ divides $k(k + 1)(k + 2) ... (k + p - 3) - 1$.
Malik Talbi
For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?
Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements:
I) The unit digits of $a^3$ and $b^3$ are different.
II) $M$ is divisible by $100$.
Please indicate which of the above statements are true with certainty.
Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.
Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.
We call a positive integer a [i]shuffle[/i] number if the following hold:
(1) All digits are nonzero.
(2) The number is divisible by $11$.
(3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$.
How many $10$-digit [i]shuffle[/i] numbers are there?