Condition of an integer divisible by $3$
10744
Prove that a number is divisible by 3 iff the sum of its digits is divisible by 3
34767
Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$
7473
How am I supposed to tell if a number is divisible by $13$ (I need a shortcut)?
30261
How can I find all integers $x≠3$ such that $x−3|x^3−3$
91044
Why is a number that is divisible by $2$ and $3$, also divisible by $6$?
24053
How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
56624
Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
128306
Show that for all $n ∈ Z$, $2n^3 + 3n^2 + n$ is divisible by both 2 and 3.
27428
Divisibility test for $4$
144132
The number $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$.
100336
Find an integer having the remainders $2,3,4,5$ when divided by $3,4,5,6$, respectively.
26336
Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
29874
Not divisible by $2,3$ or $5$ but divisible by $7$
4750
Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
117761
When is $\binom{n}{k}$ divisible by $n$?
79940
$\binom{p}{i}$ divisible by $p$, with $p$ prime
85287
Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$