If gcd$(a,2)=1,$ and $p$ is an odd prime, then $(a^{p-1})-1$ is divisible by $8$.
72511
If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
130653
$p^2$ divides $ab$ and $\gcd(a,b)=1$. Show $p^2$ divides $a$ or $p^2$ divides $b$
68145
If $p$ is an odd prime and $k$ an integer with $0<k<p-1$ then $1^k + 2^k + \ldots + (p-1)^k$ is divisible by $p$
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]
105832
prove that if $\gcd(a,b)=1$ then $\gcd(a^2, b^2)=1$
75733
Divisibility of prime numbers
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
38295
If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
156641
Show that $\gcd(3n,3n+ 2) = 1$ when $n$ is odd
96535
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
12862
Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
119282
If $a_1,a_2,\ldots, a_n$ are distinct primes, and $a_1=2$, and $n>1$, then $a_1a_2\cdots a_n+1$ is of the form $4k+3$.