query: Finding $A$ and $B$ using $ A \times B = \text{lcm}\,(A,B)\times\text{hcf}\,(A,B)$
positive_passages.docid: 25174
positive_passages.text: Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$
negative_passages.docid: 60777
negative_passages.text: Prove $\, a_1 \Bbb Z \cap \dotsb \cap a_r \mathbb{Z} = {\rm lcm}(a_1, \ldots, a_r) \Bbb Z$
negative_passages.docid: 115719
negative_passages.text: Find the possible values of |A + B + C |
negative_passages.docid: 41295
negative_passages.text: Divisibility, LCM and GCD for fractions (rationals)
negative_passages.docid: 140440
negative_passages.text: Proving $(A\times C)\cup(B\times D)\subseteq(A\cup B)\times(C\cup D)$
negative_passages.docid: 116205
negative_passages.text: System of equations in a,b,c,d
negative_passages.docid: 2626
negative_passages.text: Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$
negative_passages.docid: 99376
negative_passages.text: Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$.
negative_passages.docid: 1712
negative_passages.text: $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
negative_passages.docid: 93686
negative_passages.text: In a UFD, lcm = product for coprimes: $\,(a,b)=1,\ a,b\mid c\Rightarrow ab\mid c$
negative_passages.docid: 76652
negative_passages.text: Finding the maximum $lcm$ of a set of numbers with sum $n$
negative_passages.docid: 80684
negative_passages.text: Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$
negative_passages.docid: 28330
negative_passages.text: Finding x in $a^{x} \bmod  b = c$ when values a,b, and c are known?
negative_passages.docid: 48452
negative_passages.text: Proof that $\gcd$ divides $\operatorname{lcm}$
negative_passages.docid: 26861
negative_passages.text: $a+b=c \times d$ and $a\times b = c + d$
negative_passages.docid: 125737
negative_passages.text: GCD( LCM(a,b) , LCM(a,c) ) = LCM(b, GCD(b,c)) , why?
negative_passages.docid: 6410
negative_passages.text: Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]