query: Prove that $(a,b)^2=(a^2,b^2,ab)$
positive_passages.docid: 107914
positive_passages.text: Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$
negative_passages.docid: 104124
negative_passages.text: Prove or disprove: $A-(B-C) = (A-B)-C$
negative_passages.docid: 72092
negative_passages.text: If $A^2 = I$, $B^2=I$ and $(AB)^2=I$, then $AB = BA$
negative_passages.docid: 38200
negative_passages.text: Show that $(\mathbf{a}·\mathbf{a})(\mathbf{b}·\mathbf{b})-(\mathbf{a}·\mathbf{b})^2=(\mathbf{a}\times\mathbf{b})·(\mathbf{a}\times\mathbf{b})$?
negative_passages.docid: 81794
negative_passages.text: Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$
negative_passages.docid: 158308
negative_passages.text: Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
negative_passages.docid: 17100
negative_passages.text: Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
negative_passages.docid: 113168
negative_passages.text: If $a > b$, is $a^2 > b^2$?
negative_passages.docid: 49548
negative_passages.text: If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
negative_passages.docid: 111847
negative_passages.text: For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
negative_passages.docid: 82777
negative_passages.text: If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
negative_passages.docid: 10348
negative_passages.text: Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
negative_passages.docid: 88995
negative_passages.text: Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
negative_passages.docid: 86378
negative_passages.text: Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
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: 23080
negative_passages.text: Prove that $a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc.$