If $a^{2} + b^{2} = c^{2}$ then show that $ 3 | ab $
125210
proof: primitive pythagorean triple, a or b has to be divisible by 3
88995
Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
124868
Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
90670
Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
82777
If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
6777
Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
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 $a=b$
7491
How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?
128387
Given that $a+b\sqrt[3]{2} +c\sqrt[3]{4} =0$, where $a,b,c$ are integers. Show $a=b=c=0$
74222
If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$
87397
Show that $a^2 \bmod b = (a \bmod b)^2 \bmod b$
139082
If $a,b,c>0$ and $a^3+b^3=c^3$ then prove that $a^2+b^2-c^2>6(c-a)(c-b)$.
138707
show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$.
39779
If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.
23080
Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$
90916
Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
118981
Proving $\sqrt{ab} = \sqrt a\sqrt b$