If $m= a_1x+b_1y$, $n= a_2x + b_2y$ and $a_1b_2-a_2b_1=1$ then prove that $(m,n)=(x,y)$
25825
Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
97656
Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
133699
Proving that there is a bijection between $A$ and $B$.
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
133480
new addition and new multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z, prove the set Z equipped with these 2 new operation
24098
Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
62474
If $AB=I_n$ and $BA=I_m$ then $n=m$.
74222
If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$
16774
If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
13026
Prove that if AB=I_m, then rk(A)=rk(B).
88995
Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
20878
If $(a,b)=1$ then prove $(a+b, ab)=1$.
4141
Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$