query: If $d_1 =1, d_2, \cdots, d_k = n$ are positive divisors of natural $n$, show $(d_1d_2 \cdots d_k)^2 = n^k$.
positive_passages.docid: 27449
positive_passages.text: Product of Divisors of some $n$ proof
negative_passages.docid: 95061
negative_passages.text: Proving that $2^{2\cdot 3^{n-1}}\equiv 1+3^n\pmod{3^{n+1}}$ for every natural $n$
negative_passages.docid: 78852
negative_passages.text: If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
negative_passages.docid: 23590
negative_passages.text: Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$
negative_passages.docid: 112031
negative_passages.text: Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
negative_passages.docid: 5708
negative_passages.text: Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
negative_passages.docid: 33903
negative_passages.text: Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
negative_passages.docid: 98352
negative_passages.text: If $k|n$, then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$
negative_passages.docid: 156966
negative_passages.text: Prove that if $b = dk$ where $gcd(m, a) = d$, there are $d$ solutions to $ax \equiv b \mod m$
negative_passages.docid: 130653
negative_passages.text: $p^2$ divides $ab$ and $\gcd(a,b)=1$. Show $p^2$ divides $a$ or $p^2$ divides $b$
negative_passages.docid: 6453
negative_passages.text: Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.
negative_passages.docid: 121706
negative_passages.text: Show $\text{det}(A)$ is divisible by $2^{n-1}$
negative_passages.docid: 88995
negative_passages.text: Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
negative_passages.docid: 141909
negative_passages.text: Prove ${2}^{n}\cdot(2m+1)-1 $ is invertible
negative_passages.docid: 85688
negative_passages.text: How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 103854
negative_passages.text: Showing that $Q_n=D_n+D_{n-1}$