I was asked to prove that $1$ divides every integer and that every integer divides $0$.
I wanted some help to see how others would answer this to improve my own answer. I also wanted to ask if anyone had suggestions to help me improve my solution.
Answer: $a = qb$.
The formula above determines if $a$ is divisible by $b$ for some integer $q$.
Prove that $1$ divides every integer: Since $1$ is a factor of every integer, $1$ divides every integer. a = q \cdot 1$. So $a = q$
Since $q$=$a$, for any integer $q$, $a$ can be any integer.
Since $a$ can be any integer, by $((a / 1) = q)$, $1$ can divide any integer.
Prove every integer divides $0$. $a = qb$. $(0 / b) = q (Assuming “b” is not equal to 0)$. $0 = q$.
Since $q$ is equal to $0$ for any integer $q$, $q$ has to be equal $0$ for any integer of $b$ that is not $0$.
Since $q$ is equal to $0$ for any integer $q$ except $0$, every integer except $0$, can divide $0$.