Divisibility of Integers

When we talked about division of integers, we assumed that result of the division is an integer number.

From another side, when we talked about division with remainder, we discovered that result of the division is not always an integer number.

$a$ is divisible by $b$ if $\frac{{a}}{{b}}$ is an integer number.

Notation: ${b}{\mid}{a}$.

Alternatively, ${a}$ is divisible by ${b}$ if remainder after division equals 0.

Example 1. Determine whether 12 is divisible by 2.

12 is divisible by 2, because $\frac{{12}}{{2}}={6}$ and 6 is integer number.

Another example.

Example 2. Determine whether 15 is divisible by 4.

15 is not divisible by 4, because ${15}={4}\cdot{3}+{3}$ and remainder is not 0.

Now, do a couple of exercises.

Exercise 1. Determine whether 70 is divisible by 4.

70 is not divisible by 4, because ${70}={4}\cdot{17}+{2}$ and remainder is not 0.

Another exercise.

Exercise 2. Determine whether 100 is divisible by 5.

100 is divisible by 5, because $\frac{{100}}{{5}}={20}$ and 20 is integer number.