Found problems: 408
If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.
Let $n$ be a positive number. Prove that there exists an integer $N =\overline{m_1m_2...m_n}$ with $m_i \in \{1, 2\}$ which is divisible by $2^n$.
Prove that for every positive integer $n$, the number
$$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$
is a multiple of $8$.
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$.
(Walther Janous)
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?
Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.
Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.
Show that the number $11^{100}-1$ is both divisible by $6000$ and its last four decimal digits are $6000$.
Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$
Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.
How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$.
Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .
Let $a,b,c,d$ be integers. Prove that $ 12$ divides $ (a-b) (a-c) (a-d) (b- c) (b-d) (c-d)$.
Given is a nonzero integer $k$.
Prove that equation $k =\frac{x^2 - xy + 2y^2}{x + y}$ has an odd number of ordered integer pairs $(x, y)$ just when $k$ is divisible by seven.
Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.
Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$
Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
Prove that there are infinitely many natural numbers $ n $ such that $ 2 ^ n + 3 ^ n $ is divisible by $ n $.
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$.
Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square.
Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.
A natural number is written in each box of the $13 \times 13$ grid area. Prove that you can choose $2$ rows and $4$ columns such that the sum of the numbers written at their $8$ intersections is divisible by $8$.
How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?