# Divisibility by 8

This is the seventh post in the Divisibility Rules Series.  In this post, we will discuss divisibility  by 8.

A number is divisible by $8$ if the last three digits is divisible by $8$. For example, $25816$ is divisible by $8$ since $816$ is divisible by 8. On the other hand, $5780$ is not divisible by $8$ since $780$ is not divisible by $8$. Why is this so?

Let us start with $25 816$. First, we know that $1000$ is divisible by $8$. Therefore, $2000$, $3000$, $4000$, and all multiples of $1000$ are divisible by $8$. Since $25 816 = 25000 + 816$ and $25 000$ is divisible by $8$, we just have examine the last three digits. Notice that this is similar to $5780$. Since $5780 = 5000 + 780$, and $5000$ is divisible by $8$, we are sure that it is not divisible by $8$ since the last three digits is not divisible by $8$.

This observation can be generalized because all numbers greater than $1000$ can be expressed as multiple of 1000 + three-digit number (the hundreds, tens, and ones). Since all multiples of $1000$ are divisible by $8$, we just have to examine the divisibility of the last three digit number. Of course this observation is also similar with negative numbers. All negative numbers less than $-1000$ can can be expressed as multiple of -1000 + three-digit negative number.