What does $a\mid p$ mean?

Question

For all a, m, p, q, x Є Z such that a|p, There exists r Є W, such that 0 ≤ r < m

Such that

(px+q) MOD (ma) = q MOD (ma)     iff (ma)|p 
q MOD a + ra    iff p MOD (ma) ≠ 0

I am not able to understand this problem properly. I know that Z is a set of whole numbers and W is a set of natural numbers (Correct me if I am wrong). Specifically, I am not understanding meaning of a|p. Can anybody help me with it? Similarly, I have problems with ma|p as well.

Answer 1

$a\mid p$ means that $p$ is divisible by $a$ or $p=ak$ for some integer $k$.

Answer 2

It probably means $a$ divides $p$ ($p \bmod a = 0$)

Answer 3

This notation "|" means "divides". Don't be tricked when you see that because it does not simply mean that a is divisible by p or p is divisible by a.

For a|p to be true, it means that a and p are integers such that there is an integer c such that p=a*c

In order for a to divide p evenly, there must be a remainder of zero.

Answer 4

$a\mid p$ means $a$ divides $p$, and that means you can multiply $a$ by another integer and get $p$.

For example $4$ divides $12$ because if you multiply $4$ by some other integer (in this case $3$) you get $12$. $5$ does not divide $12$ because you can't multiply $5$ by another integer and get $12$.