How can I prove that for modular multiplicative inverses that if $∃ $[$b$]$ ∈ℤ_n$ such that [$a$] * [$b$] = [$1$], then gcd($a,n$) =$1$?
2904
Solving linear congruence (modular inverse or fraction) via gcd Bezout equation
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
34330
How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
34738
Prove that $\gcd(n!+1,(n+1)!+1)=1$
15168
If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
380
Bijection from modular multiplication?
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
14693
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
99815
Dividing versus multiplying by inverse in modular arithmetic
108450
Prove that $\gcd(a, b) = \gcd(a, b + ma)$?
154959
Proof of associativity of GCD?
97469
How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
82925
Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?