These are few questions I encountered while self teaching some elementary number theory, I've answered some but I don't have any solutions to check If I'm right.
If $a \mid m$ and $b \mid n$, prove that $ab \mid mn$.
Does $6 \mid 11n$ imply that $6 \mid n$? Justify. Does $6 \mid 34n$ imply $6 \mid n$? Justify.
If $p$ is a prime, prove that $(p, a) = 1$ or $(p, a) = p$.
If a is a positive integer, what is $(a, 2a)$? What is $(a, a2)$?, $(a, a + 1)$?, and $(a, a + 2)$?
Show that the difference of two consecutive cubes is never divisible by $3$.
Show that if $(a, b) = 1$, then $(a + b, a - b) = 1$ or $2$.
Prove that square of an integer is either a multiple of $4$ or one more than a multiple of $4$.