For $a,b,c,d\in \Bbb N$ prove that $(a-b)(c-a)(d-a)(c-b)(d-b)(d-c)$ is divisible by $12$
49501
Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$
121156
Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$
77547
Find integers $a,b,c,d$ such that
108746
Prove that there exist two integers such that i - j is divisible by n.
98963
Prove that neither $A$ nor $B$ is divisible by $5$
150528
Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.
11216
Induction: How to prove that $ab^n+cn+d$ is divisible by $m$.
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
65280
If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.
9302
Questions about Divisibility of $2^n$ by $3$
100305
Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff...
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
22845
Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
117761
When is $\binom{n}{k}$ divisible by $n$?
106416
If $n,m \in \mathbb{N}$ then there are $c,d$ such that $cd = (m,n)$, $(c,d) = 1$ and $(m/c,n/d) = 1$.
66578
Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.