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How to show that $$\gcd \Biggl( {m \choose k} , {m+1 \choose k } , {m+2 \choose k}, \cdots, {m+k \choose k} \Biggr)=1$$

where $m,k \in \mathbb{N}$ and $m \geq k $.

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    $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ is useful here.2010-10-20
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    @J. M. very clever2010-10-20

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HINT $\ \ $ if $\rm\ \ f_{k}(m+1)\ \ mod \ \ f_{k}(m)\ =\ f_{k-1}(m)\ \ $

then $\rm\ \ \gcd(f_k(m),\ f_k(m+1),\:\cdots,\:f_k(m+k)) $

$\rm\quad =\ \ \gcd(f_k(m),\ \ f_{k-1}(m),\ \cdots,\ f_0(m))\ \ \ $ [$\rm\ = 1\ $ if $\rm\ f_0(m) = 1\:$ ]