query: Show that if $(a, b) = 1$, $a|c$ and $b|c$, then $(a · b)|c$.
positive_passages.docid: 1991
positive_passages.text: If $\gcd(a,b)=1$ and $a$ and $b$ divide $c$, then so does $ab$
negative_passages.docid: 41080
negative_passages.text: If A, B, and C are set and $A \subset B ,B \subset C \rightarrow A \subset C$
negative_passages.docid: 127891
negative_passages.text: If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
negative_passages.docid: 2019
negative_passages.text: Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
negative_passages.docid: 20878
negative_passages.text: If $(a,b)=1$ then prove $(a+b, ab)=1$.
negative_passages.docid: 101362
negative_passages.text: Let a,b,c be positive real number, proof.
negative_passages.docid: 70311
negative_passages.text: Trying to prove (A ∩ B) − C = (A − C) ∩ (B − C)
negative_passages.docid: 158355
negative_passages.text: If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$
negative_passages.docid: 65280
negative_passages.text: If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.
negative_passages.docid: 124320
negative_passages.text: $a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
negative_passages.docid: 104124
negative_passages.text: Prove or disprove: $A-(B-C) = (A-B)-C$
negative_passages.docid: 97667
negative_passages.text: If $a<b$, $c<d$ then $a+c<b+d$?
negative_passages.docid: 112969
negative_passages.text: Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
negative_passages.docid: 124742
negative_passages.text: P(A|C)=P(A|B)*P(B|C)?
negative_passages.docid: 4141
negative_passages.text: Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
negative_passages.docid: 120094
negative_passages.text: Prove, that for any A, B, C sets, $A\times (B\setminus C) = (A\times B)\setminus (A\times C)$ is true.
negative_passages.docid: 128820
negative_passages.text: Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$