$6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$
48484
Prove that $6|a+b+c$ if and only if $6|a^3+b^3+c^3$
143543
Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
85941
Proof by induction that $n^3 - n$ is divisible by $6$
49501
Prove that : $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ divisible by $12$
25258
Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
118809
If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c
85449
Show a,b,c congruences when $a^2+b^2=c^2$
90916
Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
110256
Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
9302
Questions about Divisibility of $2^n$ by $3$
64654
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
109133
$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$
146851
Prove that $-3$ is a quadratic residue mod an odd prime $>3$ if and only if $p$ is of the form of $6n+1$
46814
$ a+b+c+d=6 , a^2+b^2+c^2+d^2=12$ $\implies$ $ 36 \leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \leq48 $
13904
$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
150528
Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.
44247
For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$