Show that $ax \equiv b (mod\ m) $ has solution iff $gcd(a,m)$ divides $b$
56803
Finding solutions to congruence equations
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
33469
Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
28021
Showing that these two definitions of $\gcd(a,b)$ are equivalent
978
Solving $ax \equiv c \pmod b$ efficiently when $a,b$ are not coprime
4141
Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
85688
How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
65687
$\gcd(a,b)\gcd(x,y)\mid ax+by$
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]
34330
How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?