Prove that you can choose $ 2 ^ {k-1} $ numbers from $2^k -1 $ integers so that the sum of them is divisible by $2^{k-1}$
39670
Use Pigeonhole to show of any set of $2^{n+1}-1$ positive integers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$.
82742
Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
68328
Infinitely many $n$ such that $2^n-n$ is divisible by a prime
87320
Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
85817
Show that in any set of $2n$ integers, there is a subset of $n$ integers whose sum is divisible by $n$.
130223
Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
28099
Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers.
120976
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
121706
Show $\text{det}(A)$ is divisible by $2^{n-1}$
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
74523
Proof by Cases involving divisibility
280
Prove by induction that $5^n - 1$ is divisible by $4$.
87379
How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction?
1409
Proving that the given set has at most $2^{n-1}$ elements
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $