Proving that $a+b+c $ is composite knowing it divides $abc$
96027
Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
154716
Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
50266
In the following figure prove that: $AK||BC$
80242
Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$
112969
Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
107173
If $a\lt{b}$ and $c\le{d}$, prove that $a+c\lt b+d$
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
24499
Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$
17100
Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
76049
Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
118809
If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c
116215
Prove that $P(A \cap B\mid C)=P(A \mid (B \cap C))P(B\mid C)$
24856
Proving sup(A + B) = sup A + sup B
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.