Divisibility by 8

March 3, 2012 GB High School Number Theory

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.

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Divisibility by 8

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