query: Binomial Formula evaluation for the problem $9 \mid 10^k - 1$ for $k \in \mathbb{N}$
positive_passages.docid: 9846
positive_passages.text: Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]
negative_passages.docid: 117761
negative_passages.text: When is $\binom{n}{k}$ divisible by $n$?
negative_passages.docid: 130223
negative_passages.text: Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
negative_passages.docid: 155769
negative_passages.text: Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
negative_passages.docid: 117380
negative_passages.text: Divisibility Rule for 9
negative_passages.docid: 8645
negative_passages.text: A formula for bit strings with no $k$ consecutive $1$'s using generating functions
negative_passages.docid: 87479
negative_passages.text: Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
negative_passages.docid: 21000
negative_passages.text: Calculating $\sum_{k=1}^nk(k!)$ combinatorially
negative_passages.docid: 90904
negative_passages.text: Combinatorial proof of an identity with squared binomial coefficients
negative_passages.docid: 110262
negative_passages.text: Finding a closed formula for $1\cdot2\cdot3\cdots k +\dots + n(n+1)(n+2)\cdots(k+n-1)$
negative_passages.docid: 111951
negative_passages.text: Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
negative_passages.docid: 5049
negative_passages.text: Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
negative_passages.docid: 105237
negative_passages.text: Alternating sum of binomial coefficients identity
negative_passages.docid: 71564
negative_passages.text: Combinatorial proof of identity involving central binomial coefficients
negative_passages.docid: 144447
negative_passages.text: Show that $504 \mid (n^9 − n^3 )$ for any integer $n$
negative_passages.docid: 3786
negative_passages.text: Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
negative_passages.docid: 70248
negative_passages.text: Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$