how to prove: every two numbers gcd can be divided by this two numbers any common divisor
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Intuition for $\,c\mid a,b \iff c\mid \gcd(a,b)$
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Prove: $\gcd(a,b) = \gcd(a, b + at)$.
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Prove that $\gcd(f(x),g(x)) = 1$.
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Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.
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Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
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Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
49109
Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$
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Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
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Greatest common divisor of two positive integers
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Concise proof that every common divisor divides GCD without Bezout's identity?
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Proof that $\gcd$ divides $\operatorname{lcm}$
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$p^2$ divides $ab$ and $\gcd(a,b)=1$. Show $p^2$ divides $a$ or $p^2$ divides $b$
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Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
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Divisibility, LCM and GCD for fractions (rationals)
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Prove that $\gcd(n!+1,(n+1)!+1)=1$
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Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
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Greatest common divisor of two relatively primes