Let $n = p_1^{a_1}p_2^{a_2} \dots p_k^{a_k}$. Show that $d(n) = (a_1+1)(a_2+1)\dots(a_k+1)$
102086
Prove a property of divisor function
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
119282
If $a_1,a_2,\ldots, a_n$ are distinct primes, and $a_1=2$, and $n>1$, then $a_1a_2\cdots a_n+1$ is of the form $4k+3$.
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
114277
Proving in Discrete Structures Problem that $2^N = \binom{N}{0} + \binom{N}{1} + \binom{N}{2} + \dots + \binom{N}{N}$
121880
How to show that $|D_{2n}| = 2n$ via the presentation?
5927
How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
135385
$0 < a_0 \leq a_1 \leq \cdots \leq a_n$, show $ p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0$ has no root in $|z| < 1$
136459
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
61868
Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
147661
Show that {$x_n$} is a convergent sequence if $d(x_{n+2}, x_{n+1})≤ rd(x_{n+1}, x_n)$
112631
Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$
151520
About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
14646
Prove $\det(kA)=k^n\det A$
48251
Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.