Let a, b and c be natural numbers. If a^2 = bc and gcd(b, c) = 1, then there exist natural numbers s and t such that b = s^2 and c = t^2 .
14602
Coprime cofactors of n'th powers are n'th powers, up to associates, for Gaussian integers & UFD
19700
Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
20003
Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
76680
Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
107095
Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$
157508
prove $lcm(a,gcd(b,c))=gcd(lcm(a,b),lcm(a,c))$
118809
If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
158308
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
124320
$a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
101519
What is the condition for example $(number^c)^b=number^{cb}$ to be true?
103147
$\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime
72424
Suppose 1 ≤ k ≤ n − 1 and gcd(k, n) = 1. Prove that gcd(n − k, n) = 1