query: Use of GCD Distributive Law in proof of Euclid's Lemma
positive_passages.docid: 11989
positive_passages.text: $ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
negative_passages.docid: 43728
negative_passages.text: Proof via Group Theory : $\mathrm{lcm}(a,b) \cdot \gcd(a,b) = |ab|$
negative_passages.docid: 139833
negative_passages.text: Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$.
negative_passages.docid: 80148
negative_passages.text: Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
negative_passages.docid: 28021
negative_passages.text: Showing that these two definitions of $\gcd(a,b)$ are equivalent
negative_passages.docid: 25825
negative_passages.text: Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
negative_passages.docid: 6012
negative_passages.text: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
negative_passages.docid: 137608
negative_passages.text: Proof of Euclid's formula for primitive Pythagorean Triples
negative_passages.docid: 77278
negative_passages.text: If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$
negative_passages.docid: 44076
negative_passages.text: Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$
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: 67660
negative_passages.text: Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
negative_passages.docid: 2289
negative_passages.text: Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
negative_passages.docid: 14693
negative_passages.text: $d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
negative_passages.docid: 46555
negative_passages.text: How to calculate GCD of Gaussian integers?
negative_passages.docid: 136064
negative_passages.text: How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
negative_passages.docid: 77064
negative_passages.text: Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$