Found problems: 387
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.
Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.
Is there a number whose digits are only $1$'s and which is divided by $1999$?
Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$
Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.
A pair $(m, n)$ of positive integers is called “[i]linked[/i]” if $m$ divides $3n + 1$ and $n$ divides $3m + 1$. If $a, b, c$ are distinct positive integers such that $(a, b)$ and $( b, c)$ are linked pairs, prove that the number $1$ belongs to the set $\{a, b, c\}$
Find the smallest possible positive integer n with the following property:
For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$.
(Gerhard J. Woeginger)
Let $n$ be a positive integer and $p > n+1$ a prime.
Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$
Decide whether there is an integer $n > 1$ with the following properties:
(a) $n$ is not a prime number.
(b) For all integers $a$, $a^n - a$ is divisible by $n$
Given two odd integers $a$ and $b$; prove that $a^3 -b^3$ is divisible by $2^n$ if and only if $a-b$ is divisible by $2^n$.
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.
Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.
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]
Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$.
Lê Anh Vinh
Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.
Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.
Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent:
a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold,
b) $n$ divides $504$.
Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.
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.
Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$.
1. Show that $F(r)$ is a positive integer for any prime $r \ne p$.
2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$.
3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.
Let $p$ be a prime number, $p> 5$. Determine the non-zero natural numbers $x$ with the property that $5p + x$ divides $5p ^ n + x ^ n$, whatever $n \in N ^ {*} $.
Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$.
Prove that $a \equiv n \equiv 1 \mod 10$ .
Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.
A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.
Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$