Found problems: 387
Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.
Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$.
Prove that if $n > 3$, then $6$ divides $ab$.
Determine the largest two-digit number $d$ with the following property:
for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.
Note The numbers $a \ne 0, b$ and $c$ need not be different.
Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$
then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.
a) Let $a, b, c$ be positive integers such that $ab + b + 1$, $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$, prove that $a = b = c$.
b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$.
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.
Show that there are infinitely many positive integers n such that $2\underbrace{555...55}_{n}3$ is divisible by $2553$.
Let $m$ and $n$ be positive integers such that $5$ divides $2^n + 3^m$. Prove that $5$ divides $2^m + 3^n$.
Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$.
(Malik Talbi)
Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.
Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.
For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$.
[i](tomoyo-jung)[/i]
A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.
Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$
Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.
Find all the pairs $a,b \in N$ such that $ab-1 |a^2 + 1$.
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.
Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other.
(An established theorem)
Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$,
and $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$.
Prove that $n^2$ divides $a_n$ for infinite $n$.
(Romania)
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)
Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$.
[Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.]
[hide=Hint]$n$ is squarefree[/hide]
Let $p$ be a prime number greater than $2$ and let $m, n$ be integers such that: $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.$$ Prove that $p$ divides $m$.
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 $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.
Let \(n > 1\) be an integer and \(p\) be a prime. Prove that if \(n|p-1\) and \(p|n^3-1\), then \(4p-3\) is a perfect square.