How to prove that if $\gcd(n,k)=1$ then $\gcd(f,nk)=\gcd(f,n)·\gcd(f,k)$ by primes
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
19700
Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
38295
If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
77278
If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$
139833
Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$.
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
28021
Showing that these two definitions of $\gcd(a,b)$ are equivalent
112031
Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
20574
Understanding the proof for $\gcd (F_n, F_{n+k}) = 1$ where $F_n$ is a Fermat number