Divisibility by 8

Introduction 

Among all the applications of mathematics, Division has always been a strict application of math; for easy numbers, one can attempt them easily; however, in the case of large numbers dividing them is difficult. So, in the number system, divisibility rules are described, which helps to find if a number is perfectly divisible or not. Hence, in this article, we will discuss one such divisibility rule of 8 and will know how to find if a number is divisible by eight or not. We will also find a list of numbers divisible by 8.

Divisibility by 8 rule

According to the divisibility by 8 rule, if the last three numbers are zero or divisible by 8, the whole number is divisible by 8. For example: if you consider numbers 9000 and 7432, both are divisible by eight as check the last three numbers. As in 9000, the last three numbers are zeros, and in the case of 7432, the last three numbers, 432, are divided by 8. Hence these numbers are divisible by eight and fulfil the rule. 

Let’s look at some of the examples of the divisibility rule of 8

Find if 8390 is divisible by 8? 

• First, check the last three numbers; it is 390.
• Now check if 390 is divisible by 8. So, after attempting the Division, we got 48.75 as the remainder, which means that 390 is not divisible by eight completely. 
• As you can conclude, the last three numbers were not divisible by eight completely, which means that 8390 is not divisible by 8.

Divisibility rule of 8 for large numbers

Though the divisibility for smaller numbers can be found easily as the size of the number increases, the divisibility becomes difficult. So, if you have a large number that needs to be divided by 8, check its last three numbers. 

For example, consider the number 3596723432 divided by 8.

A very long number makes it difficult to divide; however, look at its last three numbers, 432. And this number is divisible by 8, which means that 3596723432 is also completely divisible by 8.

 Find if 92536 is divisible by 8?

Just like the above question, check the last three numbers of a number that is once, tens, and thousand positions as you get 536 as the last three-digit number. 

After dividing 536 by 8, we get 67 as a remainder. This means that 536 is divisible by 8. 

So, after dividing 92536 by 8, you get 11567 as the remainder, which means this number is divisible by 8.

List of numbers divisible by 8

To find the list of numbers divisible by 8, you need to add 8 to get different numbers. 

For example, check out the list of numbers divisible by 8 

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, etc.

And in all these numbers there is a difference of 8. This means adding 8 in a number will give you the next number divided by 8.

Now, we have looked at the rule of dividing a number by 8. Let’s discuss some of the problems related to it. 

Show that n²-1 is divisible by eight if n is an odd positive integer.

Before directly jumping to a question, you need to write a general odd positive integer form. 

For some integers p, any odd positive number has the form (4p + 1) or (4p + 3). 

Let, n = 1+4p

n²−1 = (4p+1)² – 1 

On simplifying RHS, we’ll get, 

16p²+8p+1−1

8p(2p + 1)

n²−1 is divisible by 8

Now, let n = (3+4p)

n²−1=(4p+3)2−1

   16p²+9+24p−1

   16p²+8+24p

   8(2p²+3p+1)

   n²−1 is divisible by 8

Hence, n²−1 is divisible by eight if n is an odd positive integer.

Conclusion 

When solving math problems, the division is a difficult task; divisibility rules can help you simplify it. The divisibility rule of each number is described in the case of 8; if the last three numbers of an amount are divisible by 8, then the total number, be it any large number like 3529687, is divisible by 8. The same procedure determines if a number is divisible by 8. In addition, to find the list of numbers that can be completely divided by 8, you must add eight because the difference between two numbers on the list is 8.