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 $.
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 $.
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 $\rm\ f_0(m) = 1\:$ ]