query: Let $a$ and $b$ be positive integers such that $b^n + n$ is a multiple of $a^n + n$ for all natural numbers $n$. Prove that $a = b$.
positive_passages.docid: 18693
positive_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 1383
negative_passages.text: How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
negative_passages.docid: 123155
negative_passages.text: If $mn+1$ is a multiple of $24$, prove that $m+n$ is also a multiple of 24.
negative_passages.docid: 123195
negative_passages.text: Prove by induction that a^n+a^-n is an integer.
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: 97750
negative_passages.text: Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 109745
negative_passages.text: If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
negative_passages.docid: 134914
negative_passages.text: Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.
negative_passages.docid: 29692
negative_passages.text: Proof verification: elementary Number Theory Proof of $am+bn = 1$
negative_passages.docid: 17176
negative_passages.text: Proof that $x^n \mod b = (x \mod b)^n$
negative_passages.docid: 139082
negative_passages.text: If $a,b,c>0$ and $a^3+b^3=c^3$ then prove that $a^2+b^2-c^2>6(c-a)(c-b)$.
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: 99428
negative_passages.text: Prove that $3^{2n-1} + 2^{n+1}$ is always a multiple of $7$.
negative_passages.docid: 63931
negative_passages.text: When $ab/(a+b)$ is an integer, where $a,b$ are positive integers.
negative_passages.docid: 150958
negative_passages.text: Group theory question. If $m$ and $n$ are relatively prime, and $a$, $b$ belong to group and $a^m = b^m$ and $a^n = b^n$, how would one prove $a = b$?
negative_passages.docid: 62474
negative_passages.text: If $AB=I_n$ and $BA=I_m$ then $n=m$.