query: Prove that $\frac {(m,n){ n\choose m}}{n}$ is an integer for all integers $n\ge m\ge1$
positive_passages.docid: 97947
positive_passages.text: How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$?
negative_passages.docid: 60804
negative_passages.text: About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
negative_passages.docid: 157711
negative_passages.text: Number Theory: Prove that $m | \phi(a^m - 1)$
negative_passages.docid: 83935
negative_passages.text: Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$
negative_passages.docid: 72600
negative_passages.text: Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
negative_passages.docid: 63879
negative_passages.text: Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
negative_passages.docid: 96861
negative_passages.text: How find all ordered pairs $(m,n)$ such $\frac{n^3-1}{mn-1}$ is an integer
negative_passages.docid: 107397
negative_passages.text: Algebraic proof for the identity $\sum_{k=0}^m(-1)^{m-k}{{n \choose {k}}}= {{n-1}\choose m}$
negative_passages.docid: 14435
negative_passages.text: Show by induction: $(1+\frac{1}{n})^{n}<n$
negative_passages.docid: 61670
negative_passages.text: Proving that $(b_n) \to b$ implies $\left(\frac{1}{b_n}\right) \to \frac{1}{b}$
negative_passages.docid: 12967
negative_passages.text: Find all $n>1$ such that $\dfrac{2^n+1}{n^2}$ is an integer.
negative_passages.docid: 91947
negative_passages.text: Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
negative_passages.docid: 30630
negative_passages.text: Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
negative_passages.docid: 131771
negative_passages.text: Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
negative_passages.docid: 20691
negative_passages.text: prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
negative_passages.docid: 14197
negative_passages.text: show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
negative_passages.docid: 94291
negative_passages.text: $\frac{2^{2n}}{2n} \le {2n \choose n}$?