query: Show that if $a$ and $b$ are positive integers then $(a, b) = (a + b, [a, b])$.
positive_passages.docid: 142693
positive_passages.text: $\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 143130
negative_passages.text: Proving that the supremum of $(a,b)$ is $b$.
negative_passages.docid: 138707
negative_passages.text: show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$.
negative_passages.docid: 19457
negative_passages.text: Show that: $\inf(A+B) = \inf(A)+ \inf(B)$
negative_passages.docid: 151434
negative_passages.text: If integers $a$ and $b$ are not both zero, then $\gcd(a,b)=\gcd(a-b,b)$
negative_passages.docid: 98496
negative_passages.text: Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ iff $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$
negative_passages.docid: 109745
negative_passages.text: If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
negative_passages.docid: 96217
negative_passages.text: If a and b are non-negative real numbers then demonstrate inequality
negative_passages.docid: 49578
negative_passages.text: Proving $a\mathbb{Z}\cap b\mathbb{Z}=[a,b]\mathbb{Z}$
negative_passages.docid: 13026
negative_passages.text: Prove that if AB=I_m, then rk(A)=rk(B).
negative_passages.docid: 81296
negative_passages.text: If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
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: 6096
negative_passages.text: If $A+B=AB$ then $AB=BA$
negative_passages.docid: 51706
negative_passages.text: If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
negative_passages.docid: 64654
negative_passages.text: Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
negative_passages.docid: 63931
negative_passages.text: When $ab/(a+b)$ is an integer, where $a,b$ are positive integers.