Let Q(n) be the sum of digits of n. Prove that Q(n) = Q(2n) implies 9|n
90032
Problem on Sum of Digits of a Number
925
Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
149508
Prove $9^n \equiv 9 \pmod{24}\quad \forall n \in \Bbb N^*$
49687
Show that the number 9 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 9.
127833
Proving that $7^n(3n+1)-1$ is divisible by 9
28099
Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers.
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
136459
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
16660
Prove that any positive number has an $n$th root
54442
Sum of $k {n \choose k}$ is $n2^{n-1}$
153302
$9$ divides $n-r(n)$ where $r(n)$ is $n$ with its digits reversed
142666
Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$
10777
Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
92440
Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
91914
Prove that for the Fibonacci sequence $(F_n)$, $F_n$ divides $F_{2n}$.