Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $.

share|improve this question
12  
$\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ is useful here. –  J. M. Oct 20 '10 at 17:45
    
@J. M. very clever –  Ross Millikan Oct 20 '10 at 18:06

1 Answer 1

up vote 1 down vote accepted

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.