query: if $a|c$ and $a+b=c$ prove that $a|b$
positive_passages.docid: 76049
positive_passages.text: Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
negative_passages.docid: 132971
negative_passages.text: Prove that $|a| < b \iff -b<a<b$
negative_passages.docid: 152693
negative_passages.text: If $C|a$ and $C|b$ then $C|(ax+by)$
negative_passages.docid: 26766
negative_passages.text: Proving : $A \cap (B-C) = (A \cap B) - (A \cap C)$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 124742
negative_passages.text: P(A|C)=P(A|B)*P(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: 57125
negative_passages.text: Prove the distributive law $a(b+c)=ab+ac$ for real numbers?
negative_passages.docid: 151038
negative_passages.text: How to prove (A v B), (A → C), (B → D) therefore (C v D)
negative_passages.docid: 118239
negative_passages.text: If $A, B, C$ are sets prove that $(A\setminus B)\setminus C\subseteq A\setminus(B\setminus C)$
negative_passages.docid: 101159
negative_passages.text: Prove If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n)
negative_passages.docid: 60122
negative_passages.text: Prove $(1+a+b)(1+b+c)(1+c+a)\ge 9(ab+bc+ca)$
negative_passages.docid: 59239
negative_passages.text: For non-zero real $x$, $y$, $z$ and complex $a$, $b$, $c$ with $|a|=|b|=|c|$, if $x+y+z=0=ax+by+cz$, then $a=b=c$
negative_passages.docid: 25258
negative_passages.text: Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
negative_passages.docid: 110703
negative_passages.text: How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
negative_passages.docid: 5149
negative_passages.text: Show $\binom{a}{b}\binom{b}{c}=\binom{a}{a-c}\binom{a-c}{a-b}$
negative_passages.docid: 2019
negative_passages.text: Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$