query: Show that $6^{(k+1)+1} + 7^{2(k+1)-1}$ is a multiple of $43$
positive_passages.docid: 17194
positive_passages.text: $6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$
negative_passages.docid: 87379
negative_passages.text: How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction?
negative_passages.docid: 156032
negative_passages.text: Find a such that $112_a$, $63_a$ and $34_a$ are in an arithmetic progression
negative_passages.docid: 46814
negative_passages.text: $ a+b+c+d=6 , a^2+b^2+c^2+d^2=12$ $\implies$ $ 36 \leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \leq48 $
negative_passages.docid: 122006
negative_passages.text: Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
negative_passages.docid: 33177
negative_passages.text: All numbers of form $10^{k} + 1$ are composite for $k{\gt}2$ proof
negative_passages.docid: 149249
negative_passages.text: Beginner: How to complete the induction case in a proof that all multiples are a product of the least common multiple
negative_passages.docid: 126888
negative_passages.text: How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
negative_passages.docid: 10917
negative_passages.text: Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
negative_passages.docid: 66861
negative_passages.text: Show that $11^{10} \equiv 1 (\mod 100)$
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 28749
negative_passages.text: Proving that any common multiple of two numbers is a multiple of their least common multiple
negative_passages.docid: 133181
negative_passages.text: Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$
negative_passages.docid: 96535
negative_passages.text: Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
negative_passages.docid: 11612
negative_passages.text: How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
negative_passages.docid: 125040
negative_passages.text: Show by induction : $n^7-n$ is a multiple of 7
negative_passages.docid: 115145
negative_passages.text: Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits