query: prove $a c^{-1} \equiv b d^{-1} \bmod m$
positive_passages.docid: 16700
positive_passages.text: Modular reduction of fractions
negative_passages.docid: 76100
negative_passages.text: Showing that $\left\langle a,b \mid abab^{-1}\right\rangle \cong \pi_1(K) \cong \left\langle c,d \mid c^2 d^2 \right\rangle$
negative_passages.docid: 29204
negative_passages.text: $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
negative_passages.docid: 76188
negative_passages.text: Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$
negative_passages.docid: 98855
negative_passages.text: Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
negative_passages.docid: 76610
negative_passages.text: Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
negative_passages.docid: 75863
negative_passages.text: how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$
negative_passages.docid: 135986
negative_passages.text: Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
negative_passages.docid: 108237
negative_passages.text: Show $a^p \equiv b^p \mod p^2$
negative_passages.docid: 6012
negative_passages.text: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
negative_passages.docid: 114346
negative_passages.text: Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$?
negative_passages.docid: 100571
negative_passages.text: Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$
negative_passages.docid: 20161
negative_passages.text: Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
negative_passages.docid: 111515
negative_passages.text: Proving that $-a=(-1)\cdot a$
negative_passages.docid: 92630
negative_passages.text: How to prove $\gcd(a^m,b^m) = \gcd(a,b)^m$ using Bézout's Lemma
negative_passages.docid: 68705
negative_passages.text: Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
negative_passages.docid: 93820
negative_passages.text: Fermat's little theorem applied for $n$'th root existence $\!\bmod p$