$x^{13}+x+90$ is divisible by $x^2-x+a$ $(a\in\mathbb N)$. Find $a$
81226
Find all integers $a$ for which $x^2-x+a$ divide $x^{13}+x+90$.
8366
Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd
108382
Prove $x^n-1$ is divisible by $x-1$ by induction
122019
Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
104965
Find $n$ such that polynomial is divisible
156032
Find a such that $112_a$, $63_a$ and $34_a$ are in an arithmetic progression
77200
Prove:that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014
29545
$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$
105442
Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$
113994
Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$.
66578
Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
39716
Prove that $8(5)^n + 6(3)^{2n} - 80n - 14$ is divisible by $512$ for all $n \in \mathbb{N}$ by induction.
16382
Find $a$ such that $(x+a)(x+1991)+1=(x+b)(x+c)$ with $a,b,c\in\Bbb Z$
49501
Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
26940
$2^a +1$ is not divisible by $2^b-1$.
21439
For which $k\in \Bbb N$ does $x^2+x+1$ divide $ (x + 1)^{2k}+x^{2k} + 1$?