Found problems: 387
Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.
Benedek wrote the following $300 $ statements on a piece of paper.
$2 | 1!$
$3 | 1! \,\,\, 3 | 2!$
$4 | 1! \,\,\, 4 | 2! \,\,\, 4 | 3!$
$5 | 1! \,\,\, 5 | 2! \,\,\, 5 | 3! \,\,\, 5 | 4!$
$...$
$24 | 1! \,\,\, 24 | 2! \,\,\, 24 | 3! \,\,\, 24 | 4! \,\,\, · · · \,\,\, 24 | 23!$
$25 | 1! \,\,\, 25 | 2! \,\,\, 25 | 3! \,\,\, 25 | 4! \,\,\, · · · \,\,\, 25 | 23! \,\,\, 25 | 24!$
How many true statements did Benedek write down?
The symbol | denotes divisibility, e.g. $6 | 4!$ means that $6$ is a divisor of number $4!$.
Prove that if $a$ is an integer relatively prime with $35$ then $(a^4 - 1)(a^4 + 15a^2 + 1) \equiv 0$ mod $35$.
Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.
Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.
Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.
Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.
Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$.
(A Spivak)
Show that there are infinitely many positive integers $n$ such that $n$ has at least two prime divisors and $20^n + 16^n$ is divisible by $n^2$.
1) Let $a$ and $b$ be relatively prime positive integers. Prove that there is a positive integer $n$ such that $1 \le n \le b$ and $b$ divides $a^n - 1$.
2) Prove that there is a multiple of $7^{2010}$ of the form $99... 9$ ($n$ nines), for some positive integer $n$ not exceeding $7^{2010}$.
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$
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 $M$?
Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.
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 $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$.
Prove that $m$ is a divisor of $n$.
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers.
a) Find all possibilities for the three numbers that are not included in this number.
b) Give an example of such a number.
Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$
Do they exist natural numbers $m,x,y$ such that
$$m^2 +25 \vdots (2011^x-1007^y) ?$$
S. Finskii
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer
Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.
Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.
Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides
${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$
On the squares $a1, a2,... , a8$ of a chessboard there are respectively $2^0, 2^1, ..., 2^7$ grains of oat, on the squares $b8, b7,..., b1$ respectively $2^8, 2^9, ..., 2^{15}$ grains of oat, on the squares $c1, c2,..., c8$ respectively $2^{16}, 2^{17}, ..., 2^{23}$ grains of oat etc. (so there are $2^{63}$ grains of oat on the square $h1$). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by $3$.