Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

Find the smallest positive integer $N$ satisfies : 1 . $209$│$N$ 2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )

Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$

Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$. Prove that $3^{334}$ divides $a_{2001}$. (A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)

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 the sequence an be defined by $a_0 = 2, a_1 = 15$, and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$. Show that there are infinitely many integers $k$ such that $269 | a_k$.

Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.

Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.

Determine all odd natural numbers of the form $$\frac{p + q}{p - q},$$ where $p > q$ are prime numbers.

For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$. Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.

Is there a number whose digits are only $1$'s and which is divided by $1999$?

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]

Find a subset of $\{1,2, ...,2022\}$ with maximum number of elements such that it does not have two elements $a$ and $b$ such that $a = b + d$ for some divisor $d$ of $b$.

Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.

Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .

Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$. a) What is the smallest positive integer that divides exactly two elements of $M$? b) What is the largest positive integer that divides exactly two elements of $M$?

Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included). Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$