Let $a$ and $b$ be positive integers. If $b=ak$ for some $k \gt 0$ then $2^b − 1 = (2^a)^k − 1 = (2^a − 1)m$ for some $m$.
9846
Prove $n-m \mid n^r - m^r\,$ [Factor Theorem, monomial case]
112031
Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
134857
Find $(a,b)$ if $x^2-bx+a = 0, x^2-ax+b = 0$ both have distinct positive integers roots
78736
Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$
63651
If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$.
64654
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
141102
If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
137626
If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
142980
If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$
141369
If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
24098
Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
52357
If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
139503
How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
513
If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.