Prove that if $3|a^2+b^2\Rightarrow 3|a ,\, 3|b$
94516
If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$
89184
Prove $30|(a^3b-ab^3) $
56624
Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
92692
Prove $3(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\ge xyz(x+y+z)^3$
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 $a=b$
69673
Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
50614
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
126427
Proving $\sqrt{2}+\sqrt[3]{3}$ is algebraic.
37529
If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational
104498
Show that $x^3y^3(x^3+y^3) \leq 2$
27281
How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
80009
Prove that $| |a|-|b| | \le |a-b|$
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
151038
How to prove (A v B), (A → C), (B → D) therefore (C v D)
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
20964
Proof that $n^2 < 2^n$