Let $n ∈ N, n ≥ 1$. Prove that $4^n + 6n - 1$ is divisible by $9$.
124240
Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$
97805
$n+1$ is a divisor of $\binom{2n} {n}$
108746
Prove that there exist two integers such that i - j is divisible by n.
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
112147
Proving $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
38904
Prove that $(a + b + c)^{13}$ is divisible by $abc$ if $b|a^3$, $c|b^3$ and $a|c^3$.
39716
Prove that $8(5)^n + 6(3)^{2n} - 80n - 14$ is divisible by $512$ for all $n \in \mathbb{N}$ by induction.
17194
$6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$
33011
Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
74512
How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
280
Prove by induction that $5^n - 1$ is divisible by $4$.
4077
Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number.
6561
Prove that the expression $5^{2n+1} \cdot 2^{n+2} + 3^{n+2} \cdot 2^{2n+1}$ is divisible by 19.
149508
Prove $9^n \equiv 9 \pmod{24}\quad \forall n \in \Bbb N^*$
102334
How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
122019
Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$