query: If $a^a$ divides $b^b$, then $a$ divides $b$?
positive_passages.docid: 14581
positive_passages.text: How do we prove $n^n \mid m^m \Rightarrow n \mid m$?
negative_passages.docid: 29963
negative_passages.text: $6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$
negative_passages.docid: 19046
negative_passages.text: Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
negative_passages.docid: 17032
negative_passages.text: If $ab=ba$, Prove $a^2$ commutes with $b^2$
negative_passages.docid: 17316
negative_passages.text: If $a^\phi = 1$ then the order of $a$ divides $\phi$
negative_passages.docid: 64654
negative_passages.text: Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
negative_passages.docid: 73391
negative_passages.text: Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
negative_passages.docid: 31331
negative_passages.text: $|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
negative_passages.docid: 40626
negative_passages.text: Suppose that $ a,b \in \mathbb{Z}$ If 4 divides $ a^2 + b^2$, then $a$ and $b$ are not odd.
negative_passages.docid: 110256
negative_passages.text: Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
negative_passages.docid: 100549
negative_passages.text: If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?
negative_passages.docid: 146093
negative_passages.text: Why if $n \mid m$, then $(a^n-1) \mid (a^m-1)$?
negative_passages.docid: 82097
negative_passages.text: If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?
negative_passages.docid: 7894
negative_passages.text: Does it follow that $(n!)^n$ divide $(n^2)!$
negative_passages.docid: 96027
negative_passages.text: Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
negative_passages.docid: 86030
negative_passages.text: Prove or disprove - If a divides b and b divides a does a=b
negative_passages.docid: 21006
negative_passages.text: Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?