# To show that $\gcd \Bigl( {m \choose k} , {m+1 \choose k } , {m+2 \choose k}, \cdots, {m+k \choose k} \Bigr)=1$

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. –  Guess who it is. Oct 20 '10 at 17:45
@J. M. very clever –  Ross Millikan Oct 20 '10 at 18:06

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